On the accuracy of the Timoshenko beam theory above the critical frequency: best shear coefficient

Transcription

1 On th auay of th Timoshno bam thoy abov th itial fquny: bst sha offiint J. A. Fano-Villafañ * and R. A. éndz-sánhz Instituto d Cinias Físias, Univsidad Naional Autónoma d éxio, A.P. 48-, 65 Cunavaa, olos, éxio. W obtain valus fo th sha offiint both blow and abov th itial fquny by ompaing th sults of th Timoshno bam thoy with xpimntal data publishd ntly. Th bst sults a obtaind, by a last-squa fitting, whn diffnt valus of th sha offiint a usd blow and abov th itial fquny. Kywods: Timoshno bam thoy, Timoshno sha offiint, itial fquny.. Intodution In nt yas th has bn nwd intst in th flxual vibations of bams [- 4]. Evn in th simplst as of a unifom bam, thi vibations a of gat intst sin not only is th validity of th thoy still und disussion, but also th physial paamts that should b usd within it. This is th as of th Timoshno sha offiint. This adjustmnt paamt appas in th Timoshno bam thoy, TBT, to stimat th sha fo at th oss stion of a bam in tms of th sha stain at th ntoidal axis. Until now th hav bn sval thotial studis attmpting to obtain th bst valu fo th sha offiint [5-]. Som ommonly usd valus a = 5/6, * Tl ; Fax: , addsss: jofavil@fis.unam.mx Tl ; Fax: , addsss: mndz@fis.unam.mx

2 = π /, t., fo a bam with tangula oss-stion but th is no onsnsus about its valu. Expimntal studis, on th oth hand, a sa [-4]. This is du to th fat that masumnts with fqunis abov th Bnoulli-Eul gim a ndd. Numial simulations, using finit lmnts analysis, hav also bn pfomd to alulat th sha offiint [5]. To alulat th Timoshno's sha offiint it is assumd that th oss-stional aa is plana [6]. This hypothsis is ot at low fqunis but it sms to fail at fqunis high than th itial fquny f. Aft a long dbat, it was shown that in this gim, i.. fo f > f, f bing th fquny, th TBT is still valid [7] and that two familis of nomal mods appa as doublts [7-5]. On family of ths doublts is alld th ``sond TBT sptum''. In fn [7] it was also shown that th slop Ψ along th oss-stional aa is no long onstant fo th sond TBT sptum. Thus, it is xptd that th valu of th sha offiint hangs fo fqunis high than f. A qualitativ pitu of th lassifiation of th fist and sond TBT spta, in tms of th shap of th oss-stional aa, is shown in Fig.. Following th lin of Rf. [4], in this pap w will giv a nw bnhma fo th Timoshno sha offiint that is valid not only blow but also abov th itial fquny. In th nxt stion th Timoshno's bam thoy with th diffnt valus of th sha offiint is intodud. In Stion ompaison is mad of th diffnt pditions of th TBT fo diffnt valus of th sha offiint with ntly publishd xpimnt sults [7]. Th atil nds with som bif onlusions.

3 . Timoshno bam thoy Th vtial displamnt ξ in th two-offiint Timoshno bam thoy satisfis [7] 4 EI ξ 4 ρa z I A E ξ 4 ξ ρi ξ =0, t A t 4 4 z t wh and E a th sha and Young's moduli, and A, ρ, and I a th ossstional aa, th dnsity and th sond momnt of aa, sptivly. This ondimnsional thoy pdits otly th doublts [7] but assums flat dfomations of th oss-stional aa; any oth dfomation is absobd into th sha offiint. Figu. Displamnt of th oss-stional aa fo mods with fqunis blow th itial fquny f A, fo mods blonging to th fist TBT sptum A, and fo th sond TBT sptum B.

4 Whn th bam is vibating in a nomal-mod, th spaation of vaiabls mthod an b usd, iωt ξ z, t = χ z, wh χ z is th tim-indpndnt displamnt amplitud. With th pvious ansatz th Timoshno quation an b wittn as 4 d χ 4 dz d χ ρ ω dz E ω ω χ= 0, wh is th dud modulus dfind as =, 4 E and A ω = πf = ; 5 ρi f is nown as th itial fquny. Fo a f-f bam of lngth th bounday onditions, givn by th vanishing of momnts and sha fos, an b wittn in tms of th tim-indpndnt displamnt amplitud only as [7,7,0,] d χ dz z= 0, χ = z= 0,, 6 d χ dz z= 0, = dχ dz z= 0,. 7

5 Noti that Eqs. 6 and 7 dpnd on th fquny and that thy du to th Bnoulli-Eul bounday onditions, usd in Rf. [4], whn th tms on th ight a muh small than th sptiv tms at th lft, i.. fo low fqunis. Th solution to Eq., obtaind using standad mthods fo odinay diffntial quations, is givn by, xp xp xp xp = z d z z b z a z χ 8 wh a, b,, and d a onstants dtmind by th bounday onditions and ± = ± 4 ω ω ω E. 9 Fom th pvious quations, it an b sn that th itial fquny spaats th bhavio of th solutions into two gims: whn <ω ω th a two tavlling wavs and two xponntial tms in Eq. 8 sin th is imaginay and is al; whn ω ω > all th tms in Eq. 8 psnt tavlling wavs; ± a both imaginay. A shmati bhavio of ± is shown on Fig.. Th fquny sptum an b obtaind by insting Eq. 8 into th bounday onditions 6 and 7; th nomalmod fqunis a thn obtaind whn 0 = dtp with. = P 0

6 R Im Ω Ω Ω Ω Figu. Shmati bhavio of th wav vtos ± as a funtion of th fquny ω fo th Timoshno's wav quation. Th upp and low panls ospond to th al and imaginay pat of th wav vtos, sptivly.

7 . Rsults nomal-mod fquny pntag o numb of mod Figu. Pntag o in fquny btwn th xpimntal sults of Rf. [7] and th thotial pditions of th Timoshno bam thoy. Th aluminum od has a lngth = m, hight a = 0.05 m and width b = m; th lasti onstants a = 6.9 Pa, E = 67.4 Pa and ρ = g/m. Th plot mas ospond to thos givn in Tabl. Th low 4 nomal-mod fqunis of a bam of tangula oss-stion with lngth = m, hight a = 0.05 m and width b = m, w alulatd by finding th oots of th dtminant of Eq. 0. Th lasti onstants of th bam,

8 tan fom Rf. [7], a = 6.9 Pa, E = 67.4 Pa and ρ = g/m. Sval valus, found in th litatu, fo th Timoshno sha offiint = = w usd ass I, II, III, IV, and V; ths valus a givn in Tabl. Th sults w ompad with th xpimntal sults of Rf. [7]. Th o btwn ths xpimntal sults and th thoy using two diffnt valus of and is plottd in Fig.. As an b sn in this figu, xllnt agmnt btwn th thoy and th xpimnt is found at low fqunis; on an also noti that th o gows with th fquny. Clos to f, whih vais btwn 9687 Hz and 08 Hz fo th diffnt ass potd h s Eq. 5, th o btwn th diffnt thois and th xpimnt gows up to %; th lag os a found los to f, aound mod, wh a pa in th o is obsvd. In this figu on an also obsv that th sults abov th itial fquny psnt lag os than th sults blow f. Cas Symbol Timoshno sha offiints I = 5/6 II = III = 0 ν/ ν IV = 5 ν/6 5ν V =0 ν/ ν, =5 ν/6 5ν VI =0.89 VII = 0.808, = 0.89 blow f =0.800, = VIII abov f ~ =0.8687, ~ = 0.89 Tabl. Diffnt valus of th Timoshno offiints usd to tst th TBT. Th valus usd in ass VI, VII and VIII w alulatd using last squas s txt.

9 W also alulatd th bst sha offiints using last squas in th diffnt ways: with on indpndnt offiint = = as VI; with and as indpndnt offiints as VII; and, du to th hang of gim intodud by th itial fquny, with fou offiints, and fo f < f and ~ and ~ fo f > f as VIII. As an b sn fom Fig., as VIII, th o is small than 0.5 % fo all sonans blow and abov th itial fquny. Th o is small than 0.6 % in as VII. 4. Conlusions. In this pap it has bn shown that th pditions of Timoshno s bam thoy an b vy auat not only blow but also abov th itial fquny f. This was ahivd by ompaing th thotial sults fo diffnt valus of th Timoshno sha offiint with xpimntal sults ntly publishd. Whn th two offiint Timoshno bam thoy is usd, away fom th itial fquny, th diffn btwn thoy and xpimnt is small than 0.5%. This diffn is small than 0.6% if on inluds th sults that a los to th itial fquny. Th sults stongly suggst that th valu of th sha offiint abov th itial fquny is diffnt fom th valu blow it. 5. Anowldgmnts. This wo was suppotd by DAPA-UNA und pojt IN05. RAS is indbtd to J. Flos,. onsivais, A. oals,. utiz, and A. Díaz-d-Anda, fo usful

10 disussions. JAFV anowldg th suppot of CONACyT und pojt CB- 00/ W anowldg th ind hospitality of Cnto Intnaional d Cinias A. C. fo goup mtings lbatd fquntly th. 6. Rfns [] Wang, X. Q. and So, R.. C., Timoshno bam thoy: A psptiv basd on th wav-mhanis appoah, Wav otion, 57, pp [] Jun, Z., Zhushi, R., and Na, T., Engy flow, ngy dnsity of Timoshno bam and wav mod inohn, Jounal of Sound and Vibation, 54, pp [] Eslami,. R., Finit Elmnts thods in hanis, Sping Intnational Publishing, Switzland 04. [4] Chn, W. R., Paamti Studis on Bnding of Twistd Timoshno Bams Und Complx oadings, Jounal of hanis, 8, pp. N-N6 0. [5] Kano T., On Timoshno's otion fo sha in vibating bams, Jounal of Physis D: Applid Physis, 8, pp [6] Cowp,. R., Th sha offiint in Timoshno bam thoy, Jounal of Applid hanis,, pp [7] Stphn, N.. and vinson,., A sond od bam thoy, Jounal of Sound and Vibation, 67, pp [8] Rnton, J. D., A h on th auay of Timoshno's bam thoy, Jounal of Sound and Vibation, 45, pp [9] Huthinson, J. R., Sha offiints fo Timoshno bam thoy, Jounal of Applid hanis, 68, pp [0] Stphn, N.., On a h on th auay of Timoshno's bam thoy, Jounal of Sound and Vibation, 57, pp [] Chan, K. T., ai, K. F., Stphn, N.. and Young, K., A nw mthod to dtmin th sha offiint of Timoshno bam thoy, Jounal of Sound and Vibation, 0, pp [] Hamon, R. F. S., Th influn of sha and otatoy intia on th f flxual vibation of woodn bams, Bitish Jounal of Applid Physis, 9, pp [] Rosing, H. E. and Rithi, I.., On Timoshno's otion fo sha in vibating isotopi bams, Jounal of Physis D: Applid Physis, 0, pp [4] éndz-sánhz, R. A., oals, A. and Flos, J., Expimntal h on th auay of

11 Timoshno's bam thoy, Jounal of Sound and Vibation, 79, pp [5] uttmann, F. and Wagn, W., Sha otion fatos in Timoshno's bam thoy fo abitay shapd oss-stions, Computational hanis, 7, pp [6] aff, K. F., Wav otion in Elasti Solids, Dov, Nw Yo 99. [7] Díaz-d-Anda, A., Flos, J., utiéz,., éndz-sánhz, R. A., onsivais,. and oals, A., Expimntal study of th Timoshno bam thoy pditions, Jounal of Sound and Vibation,, pp [8] Stphn, N.., Th sond fquny sptum of Timoshno bams, Jounal of Sound and Vibation, 80, pp [9] Bhashyam,. R. and Pathap,., Th sond fquny sptum of Timoshno bams, Jounal of Sound and Vibation, 76, pp [0] vinson,. and Coo, D. W., On th two fquny spta of Timoshno bams, Jounal of Sound and Vibation, 84, pp [] Nstno, V. V., A thoy of tansvs vibation of th Timoshno bam, Jounal of Applid athmatis and hanis, 57, pp [] ist, B. and aughlin, J. R., Doubl Eignvalus fo th Unifom Timoshno Bam, Applid athmatial tts, 0, pp [] Stphn, N.., Th sond sptum of Timoshno bam thoy - Futh asssssmnt, Jounal of Sound and Vibation, 9, pp [4] Bhasa, A., Elasti wavs in Timoshno bams: th 'lost and found' of an ignmod, Podings of th Royal Soity of ondon A, 465, pp [5] anvih, A., Dynamis of Timoshno bam on lina and nonlina foundation: Phas lations, signifian of th sond sptum, stability, Jounal of Sound and Vibation, 4, pp