FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED BEAMS

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1 Journal of Appl Mathatcs an Coputatonal Mchancs, (), 9- FREE VIBRATION ANAYSIS OF FNCTIONAY GRADED BEAMS Stansław Kukla, Jowta Rychlwska Insttut of Mathatcs, Czstochowa nvrsty of Tchnology Czstochowa, Polan Abstract. In ths papr fr vbraton of aally functonally gra (FG) bas consstng of two sgnts s stu. Th FG bas unr consraton ar charactrz by aally varyng cross-scton aras an/or functonally grang atral proprts. Nurcal apl a ba that s clap at both ns s prsnt. Kywors: fr vbraton probl, functonally gra bas Introucton FG bas ar charactrz by unaally or spatally varabl atral proprts. A lst of paprs on FG bas s vry tnsv. For apl n [] an analytcal soluton of a statc cantlvr functonally gra ba s ulat unr th assupton that all th lastc oul of th atral hav th sa varatons along th ba-thcknss rcton. Many paprs al wth fr an c vbraton analyss of FG bas,.g. [, ]. For aally gra bas slar probls hav bco or coplcat bcaus of th govrnng quatons wth functonal coffcnts. W can nton hr [] whr fr vbraton of ponntally gra bas s analyz. Such probls ar nvstgat also n [] by usng th Frhol ntgral quatons tho. By panng th o shaps as powr srs, th rsultng Frhol quatons wr solv. In papr [] fr vbraton an stablty analyss of aally functonally gra tapr Toshnko bas ar carr out through a fnt lnt approach. In ths contrbuton th ulaton an soluton of th vbraton probl concrns th bas n th frawork of Brnoull-Eulr thory. It s assu that th changs of th cross-sctonal ara an atral proprts n th ba sgnts hav an ponntal. Th analytcal soluton of th probl s us nurcal analyss. Th ffct of slct paratrs charactrz th syst on th fr vbraton frquncs of th clap-clap ba s nvstgat.

2 S. Kukla, J. Rychlwska. Forulaton of th probl Consr a functonally gra ba of lngth (along th rcton) consst- A, ont of ng of two sgnts, havng a non-un cross-scton ara ( ) nrta I ( ), oulus of lastcty E ( ) an atral nsty ( ) of oton of th ba s gvn by [] whr ( t) E u ( ) I( ) ( ) A( ) ρ. Th quaton u ρ () t u, s th transvrs flcton of th ba at th poston an n t t. It s assu that E ρ ( ) I( ) ( ) A( ) D D < < () whr < < an, ar th nsonlss grant paratrs, D, D. ar ral constants. Substtutng () nto quaton () an assung, that u (, t) u(, t) u, t u, t <, on obtans th govrnng quatons of th ba vbratons n th an ( ) ( ) D D u u sng a sparaton of varabls accorng to u t u t,, < () whr ( ) u (, t) ( ) cosω t, u (, t) ( ) cosω t (),,, ar th corrsponng apltu functons an ω s th gnfrquncy of th ba. Substtuton of quatons () nto quatons () yls

3 Fr vbraton analyss of functonally gra bas D D < ω ω,, () Introucng th non-nsonal coornat an nsonlss quantts ω D Ω, D D µ, quatons () can b wrttn n th,, < Ω Ω µ () Aftr so transatons w can rwrt quatons () as follows,, < Ω Ω µ () Equatons () ar coplt by bounary an contnuty contons. Th contnuty contons ar ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () an th bounary contons clap-clap ba ar ( ) ( ) () () (9)

4 S. Kukla, J. Rychlwska. Soluton of th probl In ths scton a soluton of th vbraton probl ()-(9) s prsnt. Th gnral soluton of quatons () has th ( ) ( C cosδ C snδ C coshδ C snhδ), ( ) ( C cosδ C snδ C coshδ C snhδ ), < () whr δ Ω, Ω δ, δ µ Ω, µ Ω δ () Substtutng functons () nto () an (9) w obtan a st of quatons wth unknowns C,...,C. t us ntrouc th followng notatons p k k p p k k k k [, δ ] v n [ δ, ] k k [ k, δ ] v rk [ δ, k] v [( k δ ), kδ] v nk [ kδ, ( k δ )] v [( k δ ), kδ ] v rk [ kδ,( k δ ) ] v [( kδ k)(, kδ δ )] v n k [( k δ δ )(, kδ k) ] v [( kδ k)(, k δ δ ) ] v r k [( kδ δ )(, kδ k) ] v whr [ cosδ, snδ ] k v v, [ coshδ, snhδ ] nr notatons () an atr whr [ C ] T C C... an ( ) () v, k,,,,,. w ths st of quatons can b wrttn n th Aω ( ) C () A cosδ δ snδ n n n coshδ p p p δ snhδ r r r cosδ δ snδ wcosδ w w w snδ δ cosδ wsnδ wn wn wn coshδ δ snhδ wcoshδ wp wp wp snhδ δ coshδ wsnhδ wr wr wr

5 Fr vbraton analyss of functonally gra bas A non-trval soluton to quaton () sts whn th trnant of atr A s qual to zro. Thn th frquncy quaton of th ba vbraton s Equaton () s thn solv nurcally. ( ω) t A (). Nurcal apl In ths scton w prsnt so nurcal rsults. Tabl shows th frst four non-nsonal frquncy paratrs Ω (,,, ) of th FG clap- -clap ba varous valus of paratrs, wth f. an µ. It can b obsrv that an ncras of th valu of wth f causs an ncras of th frquncy paratrs Ω (,,, ). Morovr, wth an ncras n ( s f) th frquncy paratrs Ω (,, ) ar crasng. Th frquncy paratr Ω changs non-onotonc. Dagras of frquncy paratr valus Ω (,, ) as a functons of., an µ.,.,, rspctvly, ar prsnt n Fgur. Th frst four non-nsonal fr vbraton frquncs. an µ Tabl

6 S. Kukla, J. Rychlwska 9 Ω Ω Fg. Frquncy paratr valus of. µ (sol ln), Ω Ω th frst thr os of vbraton as a functon µ. (ash ln), µ (ott ln) 9 Conclusons Th fr vbraton probl of th FG ba consstng of two sgnts ach of ponntally varyng cross scton aras an atral proprts s th subct of ths papr. For a clap-clap ba th charactrstc quaton s rv. Th nurcal apls hav shown th nflunc of grant varaton on th frquncs of th FG ba. Th propos approach can b appl to th vbraton probl of ba consstng of an arbtrary nubr of sgnts. Rfrncs [] Zhong Z., Yu T., Analytcal soluton of a cantlvr functonally gra ba, Copos. Sc. Tchnol.,, -. [] Trna M.A., Bnou A., Rfn sanwch ol vbraton of bas wth b shar pzolctrc actuators an snsors, Int. J. Coput. Struct., (9), 9-9. [] Ayogu M., Taskn V., Fr vbraton analyss of functonally gra bas wth sply support gs, Matr. Dsgn,, -. [] X.-F., Kang Y.-A., Wu J.-X., Eact frquncy quatons of fr vbraton of ponntally functonally gra bas, Appl. Acoustc,, -. [] Huang Y., X.-F., A nw approach fr vbraton of aally functonally gra bas wth non-un cross-scton, J. Soun. Vb., 9, 9-. [] Shahba A., Attarna R., Marv M.T., Halar S., Fr vbraton an stablty analyss of aally functonally gra tapr Toshnko bas wth classcal an non-classcal bounary contons, Copos. Part B Eng.,, -. [] b O.I., Karnovsky I.A., Forulas Structural Dynacs, Mc Graw-Hll,.