ON UNITARY RHEOLOGICAL APPROACH OF VIBRATION ISOLATION PASSIVE DEVICES

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1 8h Inernaional DAAAM Bali Conferene "INDUTRIAL ENGINEERING - 9- April, Tallinn, Esonia ON UNITARY RHEOLOGICAL APPROACH O VIBRATION IOLATION PAIVE DEVICE Poirnihe, A.; Nasa,.; Leopa, A.; Debelea, C. & Capaana, G. Absra: This paper deals wih advaned simulaion mehods of passive devies inended for vibraion isolaion of differen dynami sensiive ehnial sysems. This sudy supposes an elasomeri maerial used for isolaor ore. Various rheologial models are used o simulae he overall behaviour of he vibraion isolaor under differen dynami loads. A finie elemen mehod provide beer ool wih he main ondiion ha suppose an appropriae onsiuive law for base maerial. The auhors propose and underlie a uniary rheologial model useful boh for lumped, and for oninuous mass ompuaional models. The main advanage of his model resuls from he easiness of simulaion of differen basi rheologial behaviors for a single servieable implemenaion. I is provided boh individual and omparaive survey of various dynamial siuaions. Key words: vibraion, isolaion, rheology, performanes, ompuaional dynamis.. INTRODUCTION Rheologial ompuaional simulaion of behaviour of differen maerials requires dediaed numerial approahes for eah single ase or aegory of analysis. During he simulaion i is very diffiul o signifian hange he numerial model beause of he differen onsiuive differenial equaions. Hereby, one model means one appliaion. The main purpose of his sudy was o formulae a uniary model whih will be able o provide differen ypes of nown rheologial models behaviour wih he help of only a few basi seings. Thus he model unning and usage beome as simple as he basi haraerisis inpu. The major uilizaion of his omple new approah frames he area of ompuaional dynamis regarding vibraion miigaion and passive isolaion of dynami aions. In order o simulae differen ases of sho, vibraion and seismi aions agains sruures he auhors was developed a numerial implemenaion of his model ino Malab se of appliaions. I had o be menioned ha he enire analysis elusively reas he elasi model wih visous damping.. MODEL ORMULATION uh as a briefly desripion of his omple model in erms of well nown lassial rheologial formulaions his ompuaional approah are based on Voig-Kelvin model sueeded by a Zenner model, and he enire ensemble insalled in derivaion wih a Mawell model [...6]. In ig. ould be found he simplified shemai diagram aording o he basi represenaion rules for rheologial models. Thus he i suggess he elasi omponens, while he i means he damping elemens. Mawell ype par onains he suession of and. Voig- Kelvin par onains a derivaion beween and. Las par, means Zenner model, onains, and basi elemens. The noaions e and () on he ig. denoes he eernal dynami eiaion, respeively he variable independen oordinae (single direion displaemen). Main hypohesis of his sudy supposes a linear behaviour of eah elemen [7...]. 87

2 88 ig.. Comple uniary rheologial model wih linear visous and elasi behaviour Using usual mehods resuls he onsiuive differenial equaion of his model as follows = () where denoes he eernal dynami eiaion and is he linear deformaion (independen oordinae), boh in respe wih ime. Terms grouping in Eqn.() was performed aording o he derivaive order of eah variable. High order derivaive erms in Eqn.() supply he model apaiy o dignify he various and inensive dynami aions and heir sruural effes inside he simulaed par. Also i enable he dynami effes analysis abou he enire ensemble base sruure - isolaor - super sruure. hor impulsive aions (lie mehanial shos, deonaions, burs aions, e.) have o be simulaed preisely oherwise heir high order derivaives beome null and he dynami apaiy of he model ould be affeed.. COMPUTATIONAL ANALYI The onsiuive equaion - Eqn.() - enables boh dynami and inemaial inpus in order o evaluae he behaviour of he model. Thus if he inpu fores are nown hen resuls displaemen response of he sysem. Also dynami response an be evaluaes for nown displaemen laws. Classial signals in evaluaion of he model responses (e.g. pulse, sep, ramp, harmoni, random) an be suessfully used aing ino aoun he analyi formulaion of onsiuive law. Compuaional approah of model behaviour requires oninuous funions beause of wo main reasons as follows: null division avoidane and realisi simulaion. The auhors evaluaed a large area of heoreial and ompuaional responses. In his paper wo groups of diagrams were presened as follows: phase domain response (see ig.) and hyserei response (see ig.). Eah ase supposes five ypes of dynami inpus (nown eernal fores a inpu) as follows: (a) uniary inpu; (b) oninuous pulse signal; () un-reurren signal; (d) reurren signal; (e) harmoni shape eiaion. Aually, hese signals represen ransien or almos harmoni signals, whih allow evaluaing of quasi-sai flowing, weigh, sep, and harmoni responses. The inpu signals for ases (b)...(d) was based on he oninuous pulse signal generaed wih he epression = ep ) ( A δ () where () denoes inpu signal; A is he eiaion magniude; δ denoes he oal

3 ime period of pulse signal - pulse base ime; is he momen of ime for maimum value of sep magniude ( = A s ) - pulse enral ime. Hereby, he un-reurren signal () means wo idenial pulse signals wih pulse enral ime having he same value wih heir base ime. This signal an be easily assimilaed wih a oninuous realisi singular sep inpu. Also, he reurren signal (d) denoes wo idenial singular sep signals (means unreurren signal) whih appear delayed beween hem. Delay ime have o be greaer han fourfold of pulse base ime. Diagrams depied in ig. show he displaemen versus veloiy phase domain response in respe wih he five ypes of inpus previously presened. () (d) (a) (b) (e) ig.. Displaemen vs. veloiy response diagrams. Graphs onain relaive unis. (see e for deails) 89

4 (a) (e) ig.. ore vs. displaemen response diagrams. Graphs onain relaive unis. (see e for deails) (b) () Liewise wih he same inpus and haraerisi parameers i was obained fore versus displaemen diagrams presened in ig.. Qualiaive analysis of he resuls reveals a sabilized evoluion for every ransien inpu. Also, for an almos harmoni eiaion he sysem provides a shorly ransien sae followed by a sabilized regime during he analysis ime unil sop. Quasi-sai flowing regime under onsan eernal eiaion pu ino he evidene a regular evoluion owards a final sable sae. I had o be menioned ha ompuaional analysis on numerial models were performed in respe wih uniary values of he haraerisi parameers in order o produe a qualiaive poin-of-view abou model apabiliy.. CONCLUION (d) Conluding remars of his analysis frames and jusify he iniial hypohesis aording o his uniary numerial model is able o assure a grea fleibiliy and high robusness in simulaion of he large domain of lassial linear rheologial models in respe wih elasi and visous behaviour. Qualiaive sudies briefly presened in his paper reveal a good 9

5 sabiliy under differen ypes of inpus, besides enough onvergene o obain eah ime he final soluion. uure researhes will regard he non-linear haraerisis inluding boh elasi and visous omponens. Main appliaions on mehanial vibraion and sho proeive ehnial soluion analysis will be primary aen ino aoun and a harmonizaion beween numerial model and insrumenal daa will be he ne real sep o finalize his researh. 5. REERENCE [] Brau, P. Elasi ruures Analysis. Behaviour under ai and Dynami Aions, Impuls Publishing House,. [] Brau, P. Theoreial Mehanis, Impuls Publishing House, 6. [] Brau P. ruural requiremens imposed o vibraion sysems, Inl. Journal of Aousis and Vibraion, 5,,. [] Brau, P. Méhodes epérimenales pour la deerminaion des araérisques élasiques e dissipaives aus sysémes anivibrailes en aouhou, The Proeedings of he Inernaional Conferene on he Theory of Mahines and Mehanisms, 7 9, Libere, Czeh Republi, Aug. ep.,. [5] Kihong hin, Joseph K. Hammond, undamenals of ignal Proessing for ound and Vibraion Engineers, John Wiley & ons, Ld, 8. [6] Benda, J.. and Piersol, A. G., Random Daa: Analysis and Measuremen Proedures, Third Ediion, Wiley- Inersiene,. [7] Howard B. Wilson, Louis H. Turoe, David Halpern, Advaned Mahemais and Mehanis Appliaions Using Malab, CHAPMAN & HALL/CRC, A CRC Press Company,. [8] Eienne Balmes, Jean-Mihel Lelere, Visoelasi Vibraion Toolbo or Use wih MATLAB, User s Guide Version.,. [9] Piersol, A.G., Paez, T.L. Harris ho and Vibraion Handboo, ih Ediion, The MGraw-Hill Companies,. [] Nasa,. Theoreial and eperimenal researhes regarding he dynami behavior of he passive vibraion isolaion of sysems, Chaper 9 in Researh Trends in Mehanis, vol. II, Eds.: Dinel P., Chiroiu V., Toma I., Ed. Aademiei Romane, 8, pp ADDITIONAL DATA Corresponding auhor: ilviu NATAC "Dunarea de Jos" Universiy of Galai Engineering auly in Braila Comple Measuremens and Virual Insrumenaion Laboraory Calea Calarasilor 9, 87 Braila Romania snasa@ugal.ro Auhors: Aurora POTIRNICHE Adrian LEOPA Carmen DEBELEAC Gigel CAPATANA "Dunarea de Jos" Universiy of Galai Engineering auly in Braila Researh Cener for Mehanis of Mahines and Tehnologial Equipmens Calea Calarasilor 9, 87 Braila Romania s: apoirnihe@ugal.ro adrian.leopa@ugal.ro bordea@ugal.ro gapaana@ugal.ro 9