EXPERIMENTAL INVESTIGATION OF TANGENTIAL CONTACT STIFFNESS AND EQUIVALENT DAMPING

Transcription

1 Proceedings in Manufacturing Systes, Volue 7, Issue, ISSN 7-9 EXPERIMENTL INVESTIGTION OF TNGENTIL CONTCT STIFFNESS ND EQUIVLENT DMPING Iuliana PISCN,*, Tierry JNSSENS, Farid L-BENDER, Cristina PUPĂZĂ ) PD. ing., Departent of Macines and Production Systes, POLITEHNIC University of Bucarest, Bucarest, Roania ) Dr. ir., post-doctoral researcer, Departent of Mecanical Engineering, KU Leuven, Heverlee, Belgiu ) Professor Dr. ir., Departent of Mecanical Engineering, KU Leuven, Heverlee, Belgiu )ssociate Professor, Dr.-Ing., Departent of Macines and Production Systes, POLITEHNIC University of Bucarest, Roania bstract: In tis paper, te penoena of ysteretic beaviour of frictional contacts observed during reciprocating experients are discussed. Te ysteretic beaviour is deterined teoretically based on te expression of ysteresis virgin curve. On te basis of te experiental analysis, an analytical etod for deterining te energy dissipation, equivalent daping and tangential contact stiffness is proposed. For all aterial contacts te sae tendencies of te energy dissipation and equivalent daping can be observed wic are increasing wit load and displaceent and, also for te contact stiffness wic is increasing wit increasing load and weakening wit te displaceent. Key words: contact stiffness, daping, ysteresis, friction, energy dissipation.. INTRODUCTION Te study of frictional contact as been te subject of scientific researc since Coulob s ypotesis []. Frictional contact appears in various ecanical systes coonly found in acine tools applications, including gears, slides, bearings, bolted joints and oters. Te ost iportant caracteristic of friction wic occurs during te process of sliding is te ysteretic effect. Te ysteretic effect can be perceived as sticking and sliding pases being caused by te beaviour of te friction force wic acts as a nonlinear spring before sliding. Tis penoenon wic appears at te icroscopic level is called pre-sliding displaceent wit non-local eory and results fro te tangential contact stiffness between contacting bodies [,, ]. Te ysteresis caracteristic can be perceived in any ecanical systes fro various applications suc as civil, ecanical, and electrical. Te ysteresis odels can be divided into rate-independent and ratedependent odels, wic in te ecanical engineering context are referred to as static and dynaic ysteresis []. Tis paper will focus on rate-independent ysteresis fro ecanical engineering perspective. In studying te ysteretic beaviour, several pysical or ateatical odels are available in te literature. Te Masing s rules [] and teir generalizations suc as fro Fan [7], distributed-eleent odels (DEMs) [], Pisarenko s etod [9], Bouc Wen odel [], Maxwell Slip odel [, ], are exaples used in te odelling of ysteresis in ecanical structures. Hysteresis penoena influence te dynaic beaviour of acine tools ecanical structures wit oving parts, wic is not copreensively exained in te literature yet. However, in oter fields of engineering, were te ysteresis penoena occur, furter researc was conducted []. In tis paper, we propose a siple approac to odel te ysteretic effects of a frictional contact. n analytical description of ysteresis loop as been described on te basis of experiental studies. Experiental studies for flat on flat frictional contacts were concentrated on te pre-sliding zone.. NLYTICL DETERMINTION OF THE HYSTERESIS Tis section presents an analytical coputation of ysteresis loops beaviour for two bodies oving relative to eac oter (see Fig.). In Fig. a typical easured ysteresis loop is sown were x =,, is te input (displaceent aplitude) and is te output, te break-away force or te static friction force liit. Te expression of te virgin curve is given by te exponential equation []. a ( x x ) ( ) ( c f x = e ). () * Corresponding autor: Splaiul Independentei, Bucarest, Roania Tel.: ; Fax: + - ; E-ail addresses: iuliana.piscan@yaoo.co (I. Piscan), Tierry.Janssens@ec.kuleuven.be (T.Janssens), Farid.l-Bender@ec.kuleuven.be (F. l-bender), cristinapupaza@yaoo.co.uk (C. Pupăză), Fig.. Frictional contact of flat on flat connection.

2 I. Piscan et al. / Proceedings in Manufacturing Systes, Vol. 7, Iss., / Fig.. Sceatic representation of ysteresis area. Fig.. Typical easured ysteresis loop. Te odel paraeters, and a c, are used to describe and to evaluate te ysteresis beaviour. Te first paraeter, representing te saturation value of te ysteresis loop can be seen as te static friction force liit, or te break-away force. Te paraeter a c represents a easure of te curvature of te ysteresis. Te paraeter x, represents a variable wic allows to sift te starting point of te ysteresis alf towards te origin of te ysteresis. Oter odel equations are also possible []. Based on te Masing s rules te ysteresis curves can be expressed by using te equations presented in tis section. If no relative displaceent between te two contacting bodies as occurred, suc tat tere is no istory of otion, and ten te friction force follows te virgin curve, f(x) were ( x) = f ( x) wit f ( x) virg virg = ( x), x ( x), virg. () x = [ exp( a )]. () x c s can be seen in Fig., if te direction of te otion canges at x =, te friction force becoes: x + ( x) = + f. () = f, () ( ) calculated wit te forula of (x) before reversal point were displaceent aplitude range = to Substituting te ysteresis curve equation into Eq. () one obtains Eq. () ( x) = ( x ) + exp a c. () Te aount of energy dissipation in one cycle of a ysteresis loop can be calculated based on te area it encloses. Te area of te ysteresis loop is calculated by using two etods: in te first etod te expression of te virgin curve is integrated and in te second etod te expression of te outer ysteresis loop is integrated... Te deterination of te area by integrating te outer ysteresis loop expression Te area enclosed in te loop (see Fig. ) gives te energy dissipation during icro-slip per cycle wic can be obtained fro integration. Tis integration of te ysteresis curve expression gives te area of te upper part as: rea rea = ( x)dx. (7) a c a c ( a ) c = + exp. () Te area of te upper part of te ysteresis loop, above te x-axis, is te sae as te area of te lower part tus resulting in a total area of te ysteresis loop, wic is te energy dissipation, as: rea t = rea. (9) By replacing Eq. () in Eq. (9) gives: rea = + exp( ac.() t ac ac.. Te deterination of te area by integrating te virgin curve expression Te ysteresis curve (Fig.) can be converted to te virgin curve, based on te Masing s rules, by dividing one alf of te curve by in te two diensions wic results in te virgin curve (Fig.). Te area witin te virgin curve loop (Fig. ) is te difference of te integral of te virgin curve expression and its linear representation wic is te global contact stiffness expression (see equations below). rea = f ( x) dx y( x) dx. () were f(x) is te virgin curve expression and y(x) is te function described by te following expression y( x) = x. ()

3 I. Piscan et al. / Proceedings in Manufacturing Systes, Vol. 7, Iss., / ( a x) k = a exp. () l c c k = virg g x. (9) Fig.. Sceatic representation of te virgin curve area. By rearranging and integrating Eq. () one can get to Eq. (). ccording to te Masing s rules, te virgin curve is extended over bot diensions, wic eans tat te area is ultiplied by te scaling factor to te power two. rea = + exp( a ) c. () ac ac Based on te Masing s rules, te resulting friction force can be calculated for any input otion trajectory of te body. Tis kind of ysteresis is called ysteresis wit non-local eory, since every velocity reversal as to be reebered until an internal loop is closed []. Terefore, te area of te virgin curve in function of te aplitude for any distribution of points in te x and y direction can be deterined based on te upper equations rea = f x) dx i i ( y( x) dx. () i y( x) = x. () i i rea = + exp( a x) i c. ) a a c c i were i =,...,. Te equivalent daping is given by: i rea ce ω =. (7) π in wic ω is te frequency, c e is equivalent daping and i is te displaceent aplitude. s can be noticed fro te equivalent daping equation te aount of energy dissipation per cycle is not dependent on te velocity, as in viscous daping, but is dependent on te aplitude of te otion. In te literature, tis type of energy dissipation is called ysteretic daping, solid daping, or structural daping. noter alternative approac is describing functions for deterining stiffness and daping given in []. Te tangential contact stiffness can ave two expressions: te first one is te global contact stiffness (9) is te ratio between force and displaceent at eac instance and te second one, te local contact stiffness, is te derivation of te virgin curve () (see related equations below): i. EXPERIMENTL DETERMINTION OF HYSTERESIS LOOPS Experiental investigation is perfored in order to deterine te caracteristics of te frictional contact. Te easureents are perfored on a previously developed triboeter [] wit soe inor adjustents. Triboeters can be used for investigating rolling or sliding friction in dry or lubricated contacts. In tis paper, te used triboeter is for sliding friction in dry conditions. Tree aterial pairs are used for te experients: luiniu on aluiniu (luiniu lloy ); Plastic on plastic (TECVINYL PVC, grey); Steel on steel (St.7). Many experients were perfored for different noral load cases and for different aplitudes, in order to deterine te evolution of te frictional force in function of te noral load, te contact stiffness and equivalent daping. Te experients were restricted to te presliding regie... Experiental results for aluiniu on aluiniu Te experientally deterined ysteresis loop and its fit for one aterial contact pair (luinu on luinu) and for one noral load, W = 9 N, at te frequency, f = Hz is sown in Fig.. Te ysteresis curves are fitted using Eq. (). Te friction force in function of te relative displaceents, or te ysteresis loops, deterined experientally were averaged over te periods in order to eliinate noise and rando beaviour effects. Te area of te ysteresis loop is calculated by using two etods: in te first etod, te expression of te virgin curve is integrated and in te second etod te expression of te outer ysteresis loop is integrated (see related equations below). In Fig. te area as function of noral load and displaceent is presented. Te equivalent daping c ω plotted in function of te noral e load and displaceent is sowed in Fig. 7. Friction Force [N] upper virgin curve lower virgin curve data saples lower ysteresis fit upper ysteresis fit Fig.. Hysteresis in pre-sliding:luinu on aluinu, W = 9 N, f = Hz.

4 I. Piscan et al. / Proceedings in Manufacturing Systes, Vol. 7, Iss., / In Figs. and Fig. 9 te global contact stiffness and, respectively te local contact stiffness are presented, wic were deterined based on te Eq. and, respectively Eq. 9. One can notice an increase of contact stiffness wit te increasing load and a decrease wit te increasing displaceent aplitude. Contact stiffness [N/µ] rea [Nµ] Fig.. Te area of te ysteresis for all loads. Fig. 9. Local stiffness as function of noral load and displaceent... Experiental results for PVC on PVC Te results of te experients for PVC are presented in tis section. In Fig. te ysteresis loops area for PVC contacting aterials in function of noral load and displaceent is presented wic as te sae increasing trend as in te previous aterial. Fig. presents te equivalent daping in function of te noral load and displaceent. Te equivalent daping, defined by Eq. (7), is increasing wit te load and decreasing wit te displaceent aplitude. Equivalent daping c e.ω [N/µ] Fig. 7. Equivalent daping in function of noral load and displaceent. rea [Nµ] Fig.. rea in function of te noral load and displaceent. Contact stiffness [N/µ] Equivalent daping c e.ω [N/µ].. Fig.. Global stiffness as function of noral load and displaceent. Fig.. Equivalent daping in function of noral load and displaceent.

5 I. Piscan et al. / Proceedings in Manufacturing Systes, Vol. 7, Iss., / Contact stiffness [N/µ] sented. Te area is increasing wit te noral load and displaceent. Fig. 7 sows te equivalent daping for all loads in function of noral load and displaceent. Due to a iger pre-sliding distance, te equivalent daping as alost te sae value as for te oter two contacting aterials. Te local contact stiffness is sown in Fig.. Te contact stiffness as te sae increasing trend wit load as for te two contacting aterials investigated in tis paper. Fig.. Local stiffness in function of noral load and displaceent. Contact stiffness [N/µ] rea [Nµ] Fig.. Fitted ysteresis data: W = 7 N, f = Hz. Fig.. Global stiffness in function of noral load and displaceent. In Figs. and Fig. te local contact stiffness and global contact stiffness are sown. lso, for tis aterial contact, te tangential contact stiffness is increasing wit load and weakening wit displaceent aplitude... Experiental results for steel on steel Te experiental results for steel on steel prove tat te syste is not powerful enoug to perfor te friction identification in pre-sliding regie as function of te load conditions. Due to te liitation in te power of te actuator only a sall portion of te ysteresis could be identified wic is alost beaving as a pure stiffness and ardly any daping is perceived for tis aplitude of otion. Te igest daping is observed at te lowest load. In Fig., te fit of ysteresis data of te two bodies and te relative displaceent of te are plotted for one noral load W =7 N. It can be seen tat for iger loads tere is not ysteresis loop due to te stiffness in te contact wic is increasing wit te noral load. In Fig. te area enclosed in te ysteresis loop for Steel in function of noral load and displaceent is pre- Fig.. rea in function of te noral load and displaceent. Equivalent daping c e.ω [N/µ] Fig. 7. Equivalent daping in function of noral load and displaceent.

6 I. Piscan et al. / Proceedings in Manufacturing Systes, Vol. 7, Iss., / Contact stiffness [N/µ] Fig.. Local stiffness in function of noral load and displaceent.. CONCLUSIONS Tis paper as considered an existing friction odel for te analytical coputation of tangential contact stiffness and equivalent daping based on te experiental easureents of tree different contact aterial pairs in a flat on flat frictional dry contact. For tis type of configuration, te friction force in function of te displaceent and noral load was investigated. typical ysteresis beaviour was observed wic is aterial, noral load and displaceent aplitude dependent. Two analytical approaces for deterining te energy dissipation of ysteresis loops are proposed bot of te agree and lead to te sae results. Te contact stiffness and equivalent contact daping are deterined fro te virgin curve expression. For all aterial contacts te contact stiffness is increasing wit te load and weakening wit te displaceent. Te equivalent daping sows an increasing trend wit te noral load and is also increasing at lower aplitude displaceent but decreasing at iger aplitude displaceent. s future work, a teoretical odel of te tangential and noral contact stiffness of flat on flat frictional contacts will be developed based on te experiental investigation. CKNOWLEDGEMENTS: Te work as been funded by te Sectoral Operational Prograe Huan Resources Developent 7 of te Roanian Ministry of Labour, Faily and Social Protection troug te Financial greeent POSDRU//./S/ and POSDRU//./S/7. Tanks go to te Departent of Mecanical Engineering fro te K.U. Leuven, wic ade it possible to perfor te experients. REFERENCES [] C.. Coulob, Teorie des acines siples (Teory of siple acines), Meoires de Mateatique et de Pysique de l cadeie des Sciences, vol., 7, pp.. [] D.D. Rizos and S.D. Fassois, Presliding friction identification based upon te Maxwell Slip odel structure, in Caos Vol., No., June, pp.. [] Habib S. Benabdalla, Static friction coefficient of soe plastics against steel and aluinu under different contact conditions, Tribology International Vol., 7, pp. 7. [] F. l-bender, W. Syens, J. Swevers, H. Van Brussel, Teoretical analysis of te dynaic beavior of ysteresis eleents in ecanical systes, International Journal of Non-Linear Mecanics, Vol. 9,, pp [] V. Lapaert, and J, Swevers, On-line identification of ysteresis functions wit nonlocal eory, Proc., IEEE/SME Int. Conf. on dvanced Intelligent Mecatronics, Coo, Italy,, pp. 7. [] G. Masing, Eigenspannungen und verfestigung bei essing (Self stretcing and ardening for brass), in: Second International Congress for pplied Mecanics, Zuric, 9, pp. (in Geran). [7] W.R.S. Fan, Te daping properties and te eartquake response spectru of steel fraes. P.D. dissertation, Univ. of Micigan, nn rbor, Mic, 9. [] W.D. Iwan, distributed-eleent odel for ysteresis and its steady-state dynaic response, J. ppl. Mec., Vol., No., 9, pp [9] G.S. Pisarenko,Vibrations of elastic systes taking account of energy dissipation in te aterial, Tecnical Docuentary Rep, WDD TR, 9. [] T.T. Baber and Y.K. Wen, Rando vibration of ysteretic, degrading systes, J. Engrg. Mec. Div., Vol. 7, No., 9, pp [] D.D. Rizos and S.D. Fassois, Maxwell Slip Model Based Identification and Control of Systes wit Friction, Proceedings of te t IEEE Conference on Decision and Control, and te European Control Conference, pp. 7-, Seville, Spain, Deceber,. [] K. De Moerlooze, F. l-bender, On te relationsip between noral load and friction force in pre-sliding frictional contacts. Part : Experiental investigation, Wear, Vol. 9,, pp. 9.