Jourl of Appled themtcs d Computtol echcs 05 () 67-7 www.mcm.pcz.pl p-in 99-9965 DOI: 0.75/jmcm.05..7 e-in 353-0588 EFFECT OF CRACK PARAETER ON FREE VIBRATION OF THE BERNOULLI-EULER BEA Izbel Zmorsk Dwd Cekus 3 teusz r Isttute of themtcs Isttute of echcs d che Desg Foudtos Czestochow Uversty of Techology Częstochow Pold zbel.zmorsk@m.pcz.pl cekus@mpkm.pcz.pl mteuszzm9@gml.com Abstrct. The pper presets the problem of free vbrto of the ctlever Beroull-Euler bem wth crck. The bem log the legth hs vrble cross-sectol re d the crck prmeters (thckess depth d locto) udergo chges. The work cludes lytcl soluto of the vbrto problem d the results obted wth the use of FE d CATIA progrm. Keywords: Beroull-Euler bem crck free vbrto mthemtcl modellg CATIA. Itroducto The bems whose geometry d/or mterl propertes chge log ther legth re mportt for stce the desg of rcrft robot rms d tll buldgs where they re used both to reduce weght or volume d to crese stregth d stblty. The result s tht the vbrto problem of bems were d re the subject of the work of my uthors [-]. I order to solve the ler boudry problems the uthors ppled vrous methods lke the ree s fucto method [ ] the Lgrge multpler formlsm [3 ] FE [5 6] or others [7 8]. tructurl elemets c fl by crck formto. I the cse whe the crck eteds the elemet t wll produce umber of chgg prmeters such s decrese the rgdty of the system d the vbrto frequecy or cresg dmpg. These chges llow for the determto of the locto d sze of the crck usg dt collected o oe or t most few vbrto frequeces d c be used for dgostc devces to detect flure the erlest stge. Dmge lyss of mechcl systems the crcked bem s terestg ssue for my reserchers [ -7 9 0]. To determe the locto d sze of crck [6 9 0] the epermetl modl lyss ws proposed. The lyss of the fluece of crck prmeters o the bem structurl behvor c be cosdered more geerlly s the problem of optml boudry codtos [].
68 I. Zmorsk D. Cekus. r The subject of the pper s bem of lerly cross-sectol re wth symmetrc crck loclzed t oe pot of the bem s legth. A theoretcl lyss of the free vbrto problem of ths system hs bee preseted. Usg the ree s fucto method the frequecy equto d the mode shpes hve bee determed. The pper presets umercl results obted o the bss of the computtol model (FE) prepred CATIA progrm.. Formulto d theoretcl soluto of the problem The subject of terest s the ctlever bem of legth L vryg cross sectol re log the legth d crck t oe pot of the bem. The model dscussed bem s show Fgure. Fg.. A scheme of the system uder study Let us cosder the model of the bem ppromted by stepped bem wth costt geometrcl d physcl prmeters of ech segmets. Fg.. A sketch of stepped bem Accordg to the Beroull-Euler theory dfferetl equto of moto of the cosdered system s: y EI s ( t) y( t) ρa () t δ( ) s() t δ( ) m () t δ ( ) m() t δ ( ) t where: y (t) s the trsverse dsplcemet EI ρa re the fleurl rgdty d the mss per ut legth of the -th bem segmet respectvely. oreover δ( ) s the Drc s delt fucto δ'( ) s the doublet fucto the fuctos s (t) d m (t) ()
Effect of crck prmeters o free vbrtos of the Beroull-Euler bem 69 re the sher force d bedg momet ctg o the rght ed of the -th segmet. The trsverse dsplcemets fuctos y y stsfy homogeeous boudry codtos gve the form: [ y ( t) ] 0 B [ y ( t) ] 0 B 0 0 d cotuty codtos t pots... : y ( t) y ( t) () L ( t) y ( t) y L The dvdg pots re: for 0 K L for L L b L ( ) for K. EI ρ A for α > 0 re s follows: (3) α EI0 EI L c 3 ( h) 3 K K () α ρ A0 K K ρ A L (5) c( h) For the free vbrto of the bem we ssume tht jωt jωt () t e m() t e y jωt ( t) Y( ) e s ( j ). Further tkg tht EI ρa EI 0 0 0 d ω where ω s the turl EI frequecy we flly get: Y IV ( ) Y( ) ( ) δ( ) δ ( ) δ ( ) δ (6) [ Y ( ) ] 0 B [ Y ( ) ] 0 B 0 0 (7) L ( ) Y ( ) Y( ) Y ( ) Y (8) The soluto of (6) determed wth the use of the ree s fucto method [ ] hs form ( EI EI ):
I. Zmorsk D. Cekus. r 70 ( ) Y (9) Ech fucto of stsfes the ohomogeeous equto ( ) δ (0) d c be wrtte s sum H 0 (soluto of the homogeeous equto 0 d soluto of equto (0)) where ( ) [ ] c c c c s sh sh cosh s cos 3 3 0 () Fuctos ( ) stsfy the sme boudry codtos s fuctos ( ) Y t for 0K d the cosdered problem they re: for K (C-C bem) 0 for (C-F bem): 0 0 0 0 Uder these codtos the costt vlues j c K j re determed. Tkg to ccout cotuty codtos (8) (9) for K we obt system of equtos: ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) 0 0 () or the mtr form: 0 AX (3) where [ ] T K K X d [ ] A j A. The elemets of the m mtr re s follows: for ( ) ( ) ( ) ( )
for K Effect of crck prmeters o free vbrtos of the Beroull-Euler bem 7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) for 3 ( ) 3 ( ) ( ) 33 ( ) 3 ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ). The otrvl soluto of the equto (3) ests for the osgulr mtr A ts yeldg to the frequecy equto: det A 0 () () s the solved umerclly wth respect to the frequeces ω m. The mode shpes correspodg to ω m re the form of (9) for the coeffcets K K depedet o. 3. mple umercl results Usg the bove formule lgorthm d computer progrm tht wll llow for the determto of the fluece of crck prmeters o the free vbrtos of the ctlever bem wth vrble cross-sectol re c be developed. I the pper the results of umercl clcultos from CATIA progrm hve bee preseted. The obted turl frequeces of the ctlever bem wth prmeters L 500 mm 75 mm b 0 mm c 5 mm for the specfed postos d szes of crck hve bee llustrted o 3D plots (Fgs. 3-8). It llowed oe to determe how the chge of the crck prmeters (g - wdth h - depth L - dstce betwee the crck d moutg pots of the bem) ffect the frst two turl frequeces of the bem. Oe of the three prmeters s costt whle the other two udergo chges o ech grph.
7 I. Zmorsk D. Cekus. r Fg. 3. Free vbrtos frequeces for g mm Fg.. Free vbrtos frequeces for g 0 mm Fg. 5. Free vbrtos frequeces for h 5 mm
Effect of crck prmeters o free vbrtos of the Beroull-Euler bem 73 Fg. 6. Free vbrtos frequeces for h 5 mm Fg. 7. Free vbrtos frequeces for L 0 mm Fg. 8. Free vbrtos frequeces for L 35 mm
7 I. Zmorsk D. Cekus. r By lyzg the fluece of chges of the crck prmeters t c be see tht the crese of thckess depth d offset vlue of the crck from the bse cuses crese of the frst d decrese of the secod free vbrto frequeces. The thckess d depth of the crck hve eglgble effect o the turl frequeces of the system ecept whe the crck s locted 0 mm wy from the clmp. Coversely chgg the dstce of the dmge from the bse of the bem hs sgfct fluece o the frst two frequeces of the system.. Coclusos The preseted theoretcl soluto llows oe to perform the umercl reserch of ot oly the fluece of dmesol chges of the crck (or crcks) but lso ts (ther) locto o the bem o the free vbrto frequeces of the system. O the bss of the coducted umercl studes the CATIA softwre t ws foud tht the thckess d depth of the crck hve lttle effect o the system but the locto of dmge cuses sgfct chges the turl frequeces of the bem. However the smllest chge the vbrto frequeces of the system my dcte dmge. Therefore use of the bove cosdertos c develop odestructve method of the detfcto of the dmge bem elemet of y cross secto d y umber of crcks. Refereces [] Kukl. Fukcje ree ch zstosow er oogrfe r 70 Wydwctwo Poltechk Częstochowskej Częstochow 009 39. [] Kukl. Zmojsk I. Frequecy lyss of lly loded stepped bems by rees s fucto method Jourl of oud d Vbrto 007 300 03-0. [3] Posdł B. Use of Lgrge multpler formlsm to lyze free vbrto of combed dymcl systems Jourl of oud d Vbrtos 99 76() 563-57. [] Cekus D. odelowe detyfkcj model bd dymk ukłdów mechczych er oogrfe r 75 Wydwctwo Poltechk Częstochowskej Częstochow 03. [5] Zheg D.Y. Kessssoglou N.J. Free vbrto lyss of crcked bem by fte elemet method Jourl of oud d Vbrto 00 73(3) 57-75. [6] Nhv H. Jbbr. Crck detecto bem usg epermetl modl dt d fte elemet model Itertol Jourl of echcl cece 005 7 77-97. [7] Owolb.. wmds A..J. eshdr R. Crck detecto bems usg chges frequeces d mpltudes of frequecy respose fuctos Jourl of oud d Vbrto 003 65 -. [8] Rychlewsk J. A ew pproch for bucklg lyss of lly fuctolly grded bems Jourl of Appled themtcs d Computtol echcs 05 () 95-0. [9] Cekus D. Wryś P. Idetfcto of prmeters of dscrete-cotuous models AIP Cof. Proc. 68 850055 (05); http://d.do.org/0.063/.930. [0] r. Zstosowe eksperymetlej lzy modlej do wykryw uszkodzeń kostrukcj Prc dyplomow żyersk Częstochow 05. [] tudzńsk R. Pozorsk Z. Błszczuk J. Optml support system of sdwch pels Jourl of Egeerg echcs 05 0033 3-8.