Expermental determnatons of the egenmodes for composte bars made wth carbon and Kevlar-carbon fbers C M Mrţou 1, M M Stănescu 2, C O Burada 3, D Bolcu 3 and V Roşca 3 1 Faculty of Mechancs, Unversty of Craova, 165 Calea Bucurest, 200620, Craova, Romana 2 Department of Appled Mathematcs, Unversty of Craova, 13 AI Cuza, 200396, Craova, Romana 3 Faculty of Mechancs, Unversty of Craova, 165 Calea Bucurest, 200620, Craova, Romana E-mal: mrtoucosmn@yahoo.com Abstract. For modal dentfcaton, the sngle-pont exctaton method has been wdely used n modal tests and t conssts n applyng a force n a gven pont and recordng the vbratory structure response n all nterest ponts, ncludng the exctaton pont. There wll be presented the expermental recordngs for the studed bars (wth Kevlar-carbon or carbon fbers), the frequency response functon n Cartesan and polar coordnates. By usng the frequency response functons we determne the egenparameters for each bar. We present the fnal panel of the egenmodes (wth the dampng factors, egenfrequences and crtcal dampng) for each consdered bar. Usng the egenfrequency of the frst determned egenmode, the bars stffness has been determned. The presented bars can be used n practcal engneerng for: car or bus body parts, planes body parts, bullet-proof vests, renforcements for sandwch beams, and so on. 1. Introducton Many papers present varous studes regardng Kevlar or carbon-fbers studes. In ths sense, n [1] there s presented a study regardng the nterfacal bondng under bendng and unaxal compresson for sandwch structures made by carbon-fber and alumnum honeycomb wth and wthout Kevlarfber nterfacal toughenng. In [2] there s presented a research on mpacted sandwch compostes wth Kevlar/hybrd and carbon face sheets subected to dfferent temperatures. Testng was performed to determne bendng and core shear stresses, maxmum energy absorpton, and absorbng energy, moment parameter, performance parameter and compresson strength after mpact. Other studes regardng these fbers can also be found, but not lmted, n [3, 8]. The modal dentfcaton s characterzed by theoretcal and expermental procedures used to determne the system egenmodes parameters. The procedures used for tests can be dvded n two mportant bg groups: sngle-pont exctaton and multple-pont exctaton. In ths paper, startng from the theoretcal bacground of modal dentfcaton, t s establshed an expermental method used to determne the egenmodes for some composte bars wth Kevlar-carbon and carbon fbers wth epoxy resn. The epoxy resn s RESOLTECH 1050 type and ts hardener s RESOLTECH 1058 type (havng a mxed densty of 1.11 g/cm 3, mxed vscosty of 633 MPa s at 23 0 C, 5% elongaton to brea and 6% flexon to brea). There was used the sngle-pont exctaton method to determne the bars Content from ths wor may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths wor must mantan attrbuton to the author(s) and the ttle of the wor, ournal ctaton and DOI. Publshed under lcence by Ltd 1
egenmodes. Ths method has been wdely used n modal testng and t conssts n applyng a force n a gven pont and recordng the vbratory response n all nterest ponts, ncludng the exctaton pont, f needed. Although the sngle-pont exctaton requres a mnmum of equpment, t needs a laborous analyss to perform extensve result processng n order to nterpret the dynamc behavor of the structure under test. 2. The modal parameters dentfcaton procedure The theoretcal procedure used n ths paper has been prevously presented by the authors n papers le [10] and [11]. So, accordng to [9], [10] and [11], f a mechancal system s descrbed by n concentrated mass ponts onted wth elastc elements (wth the next characterstcs: s stffness and μ dampng) and can have n degrees of freedom, and f t s loaded wth an {Θ(t)} external exctaton, the moton equatons can be gven by equaton (1). [ G] v... t [ ] vt [ S] v t { ( t)}. (1) In the equaton (1) we have mared wth: [G], [Μ] and [S] the matrces of mass, dampng and stffness; {v(t)} wth ts frst and second dervatves are the dsplacement, velocty and acceleraton vectors; {Θ(t)} s the generalzed forces vector. The equaton (2) s used to determne the system response at external exctaton. Its response s processed as a sum of n modal contrbutons due to each separate degree of freedom (as descrbed n equaton (2)). N T N T { v } 1 a 1 a (2) In the equaton (2) we have mared wth: {v(ω)} s the dsplacement Fourer transform; s the egenvector and ts complex conugate order; μ s the dampng factor order; ν s the a damped natural frequency order; a and are the norm constants of egenvector; ω s the external exctaton frequency. In the expermental modal dentfcaton, the modal vectors can be replaced by two constants. Ther value can be determned wth equaton (3). and 2U, 2V a a a a, (3) Accordng to the modal dentfcaton lterature, we may nsert the system admttance parameter as the rato between the dsplacement response and the force exctaton. Ths parameter may be calculated by usng the equaton (4). 2
n n U V U V 1 1, (4) The concept of dscrete system wth concentrated mass n n materal ponts was used n the approxmatons adopted durng the mathematcal model [9, 10, 11]. In order to obtan an accurate approxmaton of the real system by the dscrete one, n must converge to nfnty. Because of expermental and processng technque and of the necessary tme for data processng, ths s mpossble. The frequences doman s lmted to a reasonable wdth n practcal applcatons, whch s obtaned by the maor resonances of the analysed equpment and the frequency doman of the applcaton goal. In these condtons, the sum from equaton (4) s reduced to several components mared n the followng wth n, too. The contrbuton of superor and nferor modes are ncluded n two correctons factors nown as resdual flexblty S (for superor modes) and nferor modal admttance -(M ω 2 ) -1 (for nferor modes) (accordng to [9, 10, 11]). Usng these notatons, the equaton 4 wll have the form le n equaton (5). n 1 U V n 1 U V S ' ' 2 1 M. (5) The modal dentfcaton of a system wth n degrees of freedom assumes determnaton of 4n modal parameters: μ, ν, U, V. These are ntrnsc characterstcs of the system, ndependent of the external condtons. 3. The modal dentfcaton for the chosen composte bars In fgure 1 there s presented the expermental setup and the analyzed composte bars (a bult-n end and the other one s free wth an accelerometer placed at the mddle of the free length). The bars are mared wth 1, 2 and 3 accordng to fgure 1. The samples have the thcness of 0,002 m, the wdth 0.05 m and the densty: 1090 g/m (sample 1), 1170 g/m (sample 2) and 1260 g/m (sample 3). Fgure 1. Expermental setup: a) the studed samples; b) a scheme wth the expermental setup. 3
In order to dentfy the egenmodes, the next steps were followed: - we have mared wth 1 and 2 the samples wth Kevlar-carbon fbers (the dfference between them s the drecton of fber applcaton, as shown n fgure 1) and wth 3 the sample wth carbon fber; - the expermental recordngs are determned (the mpact force made wth an mpact hammer wth the senstvty 1,020 pc/ms -2 and the vbratory response wth an accelerometer wth 0,04 pc/ms -2 senstvty); - the frequency response functons (abbrevated as FRF n the next paragraphs) are determned n both Cartesan and Nyqust coordnates; - the resonances approxmate poston and the ntal modal parameters μ and ν calculus (where s the number of the modal parameter); - the frst stage dentfcaton of modal parameters μ, ν, U, V, S and (M ) -1 on lmted frequency domans (the dentfcaton s made by usng lnear procedures, determnng those modal parameters that, nserted n relaton (5), generate theoretcal characterstcs whch approxmate wth mnmal error the expermentally determned frequency response functon); - the fnal stage of modal parameters μ, ν, U, V, S and (M ) -1 over the entre doman (the dentfcaton s made by usng nonlnear procedures of recursve approxmaton, determnng those modal parameters that, nserted n equaton (5), generate theoretcal characterstcs whch approxmate wth mnmal error the expermentally determned frequency response functon). The tests are performed n order to determne the bars egenmodes, egenfrequences, the dampng factors and the crtcal dampng for each egenmode. a. b. Fgure 2. Sample 1: a) Expermental recordngs; b) FRF n Cartesan coordnates. a. b. Fgure 3. Sample 1: a) Real and magnary parts of FRF; b) FRF n polar coordnates. Table 1. The modal parameters obtaned for the consdered samples. Sample Egenmode μ (mu) ν (nu) 1 [-(Ns/m)/g] [1/s] U V ζ (zta) [(Ns/m)/g] 4
2 3-2.223 111.589 40.873-26.600 0.019-5.922 636.327-312.888 71.232 0.009-11.893 1383.824 173.138 49.069 0.008-1.969 94.378 42.941-0.230 0.020-4.103 539.016-9.932 179.545 0.007-8.641 1254.753 61.749-32.395 0.006-1.698 105.322 49.348-22.632 0.016-5.461 601.732-159.533 123.668 0.009-14.943 2138.874 569.156 326.391 0.006 Fgure 4. Fnal panel of the modal parameters: a) Sample 1; b) Sample 2; c) Sample 3. Usng the relaton (6) we have determned the loss factor for each egenmode. η= 2 ζ (6) We have obtaned the next values: - sample 1: 0.038 (frst egenmode); 0.018 (second egenmode); 0.016 (thrd egenmode); - sample 2: 0.040 (frst egenmode); 0.014 (second egenmode); 0.012 (thrd egenmode); - sample 3: 0.032 (frst egenmode); 0.018 (second egenmode); 0.012 (thrd egenmode). Usng relaton (7), from [13], we have determned the bars flexural rgdty. 0.5 EI 0.5 2 A l 4.694 0,1591549 (7) 5
4. Conclusons In ths paper, the modal parameters were determned for three types of composte bars mared n ths way: sample 1 and 2 bars wth Kevlar-carbon fbers wth dfferent orentaton and sample 3 a bar wth carbon fber. All the fbers were combned wth epoxy resn. The sngle-pont exctaton method was used. For all the bars, from the expermental setup, three egenmodes were determned. The above presented experment shows many dstnct egenmodes n a large frequency doman: from 94.378 s -1 up to 2138.874 s -1. The modal dentfcaton method has the advantage that t s non-destructve and can be used n the case of complex structures bult from modern (le composte materals, shape memory alloys, and so on) or classcal (le steel, alumnum or cast ron structures) materals. But ths method also presents some dsadvantages because t can be used for some mechancal parameters calculus (for example, the ones that characterze the breaage of a certan structure). We have determned the loss factor usng the crtcal dampng value (ζ (zta)) and the bars stffness. In practcal engneerng, these types of compostes structures can be used for: car or bus body parts, planes body parts, bullet-proof vests, renforcements for sandwch beams, and so on. Acnowledgement Ths wor was supported by the strategc grant POSDRU/159/1.5/S/133255, Proect ID 133255 (2014), co-fnanced by the European Socal Fund wthn the Sectoral Operatonal Program Human Resources Development 2007-2013. References [1] Sh S, Sun Z, et al. 2014 Carbon-fber and alumnum-honeycomb sandwch compostes wth and wthout Kevlar-fber nterfacal toughenng Compostes: Part A 67 pp 102-110 [2] Saleh-Khon A and Mahnfalah M 2007 Temperature effects on Kevlar/hybrd and carbon fber composte sandwches under mpact loadng Compostes structures 78 pp 197-206 [3] Sargans J and Suhr J 2012 Core materal effect on wave number and vbratonal dampng characterstcs n carbon fber sandwch compostes Compostes Scence and Technology 72 pp 1493-1499 [4] Wong Y C, Ruan D, et al. 2014 The nfluence of magnetc feld on ballstc performance of aramd fbre and ultrahgh molecular weght polyethylene Materals and Desgn 64 pp360-365 [5] Gustn J, Joneson A, et al. 2005 Low velocty mpact of combnaton Kevlar/carbon fber sandwch compostes Composte Structures 69 pp 396-406 [6] Caserta G D, L. Ianucc, et. Al 2011 Shoc absorpton performance of a motorbe helmet wth honeycomb renforced lner Composte Structures 93 pp 2748-2759 [7] Mamvand M and Laghat G H 2010 A model for ballstc mpact on mult-layer fabrc targets Internatonal Journal of Impact Engneerng 37 pp 806-812 [8] Ong C W and Boey C W 2011 Advanced layered personnel armor Internatonal Journal of Impact Engneerng 38 pp 369-383 [9] Edwns D 1987 Modal Testng. Theory and Practce Bruel&Kaer [10] Manea I, Bolcu D and Mrţou C 2007 Software for mechancal systems modal dentfcaton Annals of the Unversty of Craova. Mechancal Seres 1 pp 191-199 [11] Mrţou C, Bolcu D, et. al 2014 Expermental Determnatons of the Egenmodes for Sandwch Bars Wth Dfferent Core Renforced wth Metal Fabrc Appled Mechancs and Materals 658 pp 255-260. [12] Mrţou C, Ilncou D, et al. 2011 A Comparson Between the Modal Parameters Obtaned by two Dfferent Accelerometers The 5 TH Internatonal Conference on Manufacturng Scence and Educaton 1 pp 39-43 [13] Burada C O and Mrţou C M, et. al 2014 Expermental determnatons of the dampng factor and stffness for new sandwch platbands wth dfferent renforcement and core Romanan Journal of Materals 44 pp 405-413 6