TRANSFER MATRIX METHOD FOR FORCED VIBRATIONS OF BARS

Transcription

1 U.P.B. Sc. B., Seres D, Vo. 7, Iss., ISSN TRANSFER MATRIX METHOD FOR FORCED VIBRATIONS OF BARS Vaentn CEAUŞU, Andre CRAIFALEANU, Crstan DRAGOMIRESCU 3 Lcrarea prezntă metoda matrceor de transfer, apcată a vbraţe forţate ae bareor drepte. Se consderă caz vbraţor axae, ae bareor c secţne varabă dscontn, acţonate de forţe pertrbatoare concentrate. Snt, de asemenea, prezentate câteva apcaţ reatv smpe. The paper presents the transfer matrx method, apped to forced vbratons of straght bars. The case of axa vbratons s consdered, of bars wth dscontnosy varabe cross-secton, acted pon by concentrated pertrbaton forces. Some reatvey smpe appcatons are aso presented. Keywords: vbraton of bars, transfer matrx.. Introdcton Compared to the fnte eement method, the transfer matrx method s sed more and more n the stdy of contnos system vbratons [], [], [6], [7]. The method s sed for ongtdna, torsona and bendng vbratons, as we as for any of ther combnatons [6], [7], [], []. Some appcatons of the method have aready been presented by the athors n references [], [3]. Ths paper stdes the forced vbratons of bars. The case of axa vbratons s consdered, of bars wth dscontnosy varabe crosssecton, acted pon by concentrated pertrbaton forces. Some reatvey smpe appcatons are aso presented.. Presentaton of the method The transfer matrx method s based on estabshng reatons between state vectors n two sectons, by means of fed matrxes. Ths, for the stdy of free axa vbratons (fg. ), the reaton s [], [7], [], [], []: Prof., Dept. of Mechancs, Unversty POLITEHNICA of Bcharest, Romana Reader, Dept. of Mechancs, Unversty POLITEHNICA of Bcharest, Romana, e-ma: ycraf@yahoo.com 3 Reader, Dept. of Mechancs, Unversty POLITEHNICA of Bcharest, Romana

2 36 Vaentn Ceaş, Andre Crafaean, Crstan Dragomresc ω c ω cos, sn, = c EA ω c, () N ω ω ω EA sn, cos N, c c c where the foowng notatons have been ntrodced: propagaton speed of ongtdna waves, E c = ; () ρ A cross-secton area; E Yong s mods; ω crcar freency of the vbraton. N N (x,t) x, Fg. By sng the anttes [], [3] ω = α, (3) c respectvey, N =, N =, (4) reaton () becomes: cosα, snα, = (). (5) sn, cos, α α N N In reatons (5), the stats vectors {} () () {} () = = N, = = N (6) N N have the dmenson of a ength. Non-dmensona notatons can be aso sed. Ths, by choosng a reference dspacement, ~, wth notatons N ~ =, ~ N =, (7) reaton (5) becomes { ~ } ( ) [ ] (, ) { ~ } = A N, (8)

3 Transfer matrx method for forced vbratons of bars 37 where fed matrx [ ] (,) ( ), has the same expresson as n reaton (5),.e. [ ] (, ) cosα =, snα A, A, aso caed transfer matrx from secton () to secton. (9) snα, cosα, By sng notatons (4) and (6), reaton (5) can be wrtten more concentrated: {} = [ A] (,) {}. () In prevos papers, [], [3] the transfer matrx method has been apped for the stdy of free ongtdna, torsona and bendng vbratons, of straght bars wth constant, dscontnosy varabe and contnosy varabe cross-secton. 3. Forced vbratons For the stdy of forced vbratons, frst the case of a bar acted pon by a concentrated pertrbaton force s consdered. The pertrbaton force s chosen n the form F = F, cos Ωt (fg. ). - F =F, cosωt x () (r) dx F =F, cosωt () N Fg. In secton, an eement of nfntesma ength dx s consdered, wth a secton to the eft of the appcaton pont of the force ( ( ) ) and one to the ( r ) rght ( ). Reatons between the stats anttes are: ( r) = () ( r) N = N F cosωt. The second reaton () can be wrtten (r) N

4 38 Vaentn Ceaş, Andre Crafaean, Crstan Dragomresc ( r) ( N N ) F, = cosωt () or ( r) ( = ) N F N cos Ω t, (3) where F, F =, (4) Ω = α. c (5) Reatons () can be wrtten, foowng the method presented n references [3] and [7], by means of a 3 3 jmp matrx, for the axa force: ( r) ( ) () r N N = cosωt F. (6) For an eement stated between sectons and, for whch the pertrbaton force Fp, = F, cos Ωt s apped n secton the foowng reaton can be wrtten { } (, r ) [ ] () [ ] (, ) S { } (, = r) F A, (7) where { } { }, r, r, r (, r) = N, = N, (8) [ ] [ ] () cosα sn α, A = sn α cosα, SF = F cosωt. (9) In the case of forced vbratons, the permanent soton (the partcar soton) mst be determned, whch, for F, cos Ωt, has the expresson: ( x, t) = X ( x) [ Acos Ωt + B sn Ωt]. () The constants A and B, as we as the fncton X ( x), can be determned accordng to the bondary condtons, by wrtng the reatons between stats matrxes at the two ends of the bar.

5 Transfer matrx method for forced vbratons of bars Appcatons Two reatvey smpe appcatons are presented, n order to strate the transfer matrx method. 4.. Bar fxed at one end, wth pertrbaton force apped at the free end In ths case (fg. 3), snce the pertrbaton force s apped at the end of the bar, t can be ntrodced by bondary condtons and the artfce presented above s not necessary. However, the method s sed as t was descrbed. F p =F cosωt x Fg. 3 Wth the notatons n Fg. 3 and from (6) and (4), t foows: () ( ) cosα sn α (, r) ( ) N = F cosωt sn α cosα N, () whch s evaent to () ( ) ( ) = cosα + N sn α (, r) ( ) ( ) N = sn α + N cosα F cosωt ( ) =. For the bar fxed at one end and free at the other, the bondary condtons are =, N = () or ( ) (, r) =, N =. ( ) System ( ) becomes, wth the nta condtons: N = snα (3) N F = cosα cosωt.

6 4 Vaentn Ceaş, Andre Crafaean, Crstan Dragomresc It foows: F N = cos Ωt cosα (4) = F tgα cosωt. Observaton. In the case cos α =, (5) respectvey, ω cos =, (5 ) c system (3) eads to, N. Ths happens de to the resonance phenomenon, for whch the partcar soton s not n the form Acos Ωt + Bsn Ωt, bt n the form A t cosωt + Bt sn Ωt. Reaton (5 ) represents the eaton of the egenfreences, for free ndamped vbratons. Ths eaton has the soton π c ωk = ( k ). (6) 4.. Bar fxed at both ends, wth pertrbaton force apped at the mdde Wth notatons n Fgre 4, t rests for the segment ( ), () α cos, N = sn α or, concentrated, sn α cosα { } (, ) [ ] (, ) A { } ( ) ( ) ( ) N, (7) =. (7 ) F p =F cosωt / / r / / F p =F cosωt Fg. 4

7 Transfer matrx method for forced vbratons of bars 4 In secton () t rests () ( ) (, r) (, ) N = F cosωt N, (8) or, concentrated, { } (, r ) [ ] ( ) S { } (, = ) F. (8 ) For segment ( ), α α cos sn (), (, r) N = sn α cosα N, (9) respectvey, { } ( ) [ A ] (, ) { } (, r) =. (9 ) From reatons (7 ), (8 ) and (9 ), t rests: { } ( ) [ ] ( ) [ ] () [ ] (, ) = A, S A { } ( ) F. (3) Reatons (3) are accompaned by the bondary condtons: ( ) (, ) = =. (3) In reatons (3) and (3), the nknowns are ( ) N, N,.e. N () t and N () t. The tme fncton can be easy dentfed as: cos Ωt. Hence: N ( t) = Nˆ Ωt N ( t) = Nˆ cos, cosωt, (3) so that ony the scaars ˆN and ˆN reman as nknowns. The eaton of the egenfreences can be aso estabshed, by eatng wth zero the determnant of the near system n ˆN and ˆN, rested from eaton (3) wth condtons (3). The matrx form of the eatons n ˆN and ˆN factates the se of compter codes, sch as MATLAB, Mathcad or even Exce, n order to sove the probem. Indeed, by choosng = m, 4 4 m A =, Ω =s, F = N,

8 4 Vaentn Ceaş, Andre Crafaean, Crstan Dragomresc kg 6 N E = 7.86, ρ =., the foowng vaes have been obtaned: 3 m m N ˆ = 5.43N and N ˆ = 5.43 N. 5. Concsons Transfer matrx method can be sed aso for sovng probems of forced vbratons of bars. By sng the artfce presented n reatons (7), the matrx form of the eatons can be easy wrtten n order to determne the tme fncton and the vaes of the nknown anttes. The exampes presented above show how the method can be apped. 6. Acknowedgement Ths stdy was spported by research contract No. 8-3 / 7 (SIMOCA), fnded by the Natona Athorty for Scentfc Research n Romana (Mnstry of Edcaton, Research and Yoth) throgh the Natona Center for Projects Management (CNMP). R E F E R E N C E S []. Gh., Bzdgan, Lca Fetc, M. Radeş, Vbraţe sstemeor mecance (Mechanca Systems Vbratons, n Romanan), Ed. Academe, Bcreşt, 975. []. V. Ceaş, A. Crafaean, Cr. Dragomresc, A. Costache, Mara Prnă, Transfer Matrx Apped to Latera Vbraton of the Beams, Second Int. Conf. of SRA on Sond and Vbraton, Bcharest, 4-7 oct. 4. [3]. V. Ceaş, N. Enesc, A. Crafaean, Cr. Dragomresc, The axa force nfence on the bars transversa vbratons, Anaee Unv. Martme Constanţa, an VII, vo. 9 (II), Constanţa, 6. [4]. W. R. Cogh, J. Penzen, Dynamcs of Strctres, McGraw-H Book Company, 975. [5]. M. Géradn, D. Rxen, Mechanca Vbratons. Theory and Appcaton to Strctra Dynamcs, Wey-Masson, 994. [6]. O. Gash, F. X. Magrans, The Goba Transfer Drect Transfer Method Apped to a Fnte Smpy Spported Eastc Beam, Jorna of Sond and Vbraton, 76, pp , 4. [7]. M. Laanne, P. Berther, J. Der Hagopan, Mécane de vbratons néares, Masson, 986. [8]. W. Nowack, Dnamca sstemeor eastce, Ed. Tehnca, Bcreşt, 969. [9]. S. Tmoshenko, D. H. Yong, W. Jr. Weaver, Vbraton Probems n Engneerng, Forth Ed., John Wey&Sons, 974. []. R. Vonea, D. Vocesc, Vbraţ mecance, (Mechanca Vbratons, n Romanan, prnted corse) I. P. B., 979. []. R. Vonea, D. Vocesc, Captoe specae de mecancă teoretcă (Speca Chapters of Theorca Mechancs, n Romanan), I.P.B., 99. []. R. Vonea, D. Vocesc, P.F. Smon, Introdcere în mecanca sod c apcaţ în ngnere (Introdcton n Sod Mechancs wth Appcatons n Engneerng, n Romanan), Ed. Academe, Bcreşt, 989.