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 stanslaw.kukla@.pcz.pl, owta.rychlwska@.pcz.pl 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.
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
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)
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
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...999....9.99..9.....9..9.....9.....9.9..9.......9.......9....9.99.99.9...9.9.9....9...9.
S. Kukla, J. Rychlwska 9 Ω Ω...... 9............ 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,.