CEE49b Chaper - Free Vbrao of M-Degree-of-Freedo Syses - I Free Udaped Vbrao The basc ype of respose of -degree-of-freedo syses s free daped vbrao Aaogos o sge degree of freedo syses he aayss of free vbrao yeds he ara freqeces of he syse For he aayss, he easc resorg properes of he syse s be descrbed frs Ths ca eher be doe ers of sffess or fexby Srcra Sffess Sffess of a srcre s descrbed by he sffess arx, whose eees are defed as he force acg a ode, order o prodce a soe dspacee a ode I ped ass odes, he sffess cosas defed above are deca o he sffess sed sac odes Exape - M-sorey shear bdg A shear bdg s oe where he ressace o aera oads s fro he bedg of he cos he foors are fey rgd ad he cos are fxed-eded where coeced o he foors I I I Fg M Sorey Shear Bdg Page -
CEE49b EI The basc sffess cosa for a co sbeced o shear oy s, where EI s he bedg sffess of he co ad s he co egh The asseby of he sffess arx s perfored oe eee a a e, wh each foor of he bdg seqeay sbeced o a shear dspacee ad he sffesses added as approprae eg he sffess eee for he frs foor of he shear bdg Fg, de o a dspacee of he frs foor s: EI EI he per s for cos per sorey The f sffess arx for he -sorey shear bdg s: EI EI EI EI EI EI EI EI EI Noe: - he arx s dagoay syerc Sac Codesao Ths s he er gve o he spfcao of a sffess arx hrogh he eao of degrees of freedo For exape, os bdgs ad srcres exposed o aera oads, here are o sgfca exera oes or ass oe of era acg he os Therefore he o roaos ca be eaed fro he goverg eqaos, so he deforao of he srcre ca be expressed ers of aera dspacees oy Cosderg he f 4x4 sffess arx for he co show beow, he eees ca be assebed oe degree of freedo a a e We sha see how hs ca be spfed sg Sac Codesao Page -
CEE49b Reca he sffess characerscs of a fxed eded bea: P P A B A B M A M A P EI P EI M B M B M A M P M A B P EI / P EI M / M B / EIθ M A M B θ P MB M A θ P P θ MB 4EI θ M A EI θ P 4 4 EI P EI Fg 5 a b a b a Geerao of f sffess arx 4x4 ad b Codesed x Page -
CEE49b Assebg he eees of he copee sffess arx, we oba: EI EI 4 4 EI EI 4 4 4EI 4EI EI 4 4 4EI 44 Now, f oy sac horzoa forces P, ac, he arx eqao reag he p forces o he op dspacees ad roaos s: P 4 P 4 4 ψ 4 4 4 44 ψ or, geeray: P A T B B Cψ oe ha he ower par of he eqao: {} [ B] T {} [ C]{ ψ} Page -4
CEE49b yeds he roaos: T { ψ } [ C] [ B] {} he sbsg hs expresso for he ow roaos ers of he ow dspacees o he pper par of he eqao: T { P} [ A]{} [ B][ C] [ B] { } T { P} [ A] [ B][ C] [ B] {} he Codesed Sffess Marx s he: T [ ] [ A] [ B][ C] [ B] I hs case, s a x arx, where s he ber of rasaoa degrees of freedo hs case a x arx vovg rasaos oy If he roaos are desred, s a spe aer o ser he resg dspacees o he ow reaoshp bewee dspacee ad roao Goverg Eqaos for he Soo o he Free Vbrao Probe Degrees-of-Freedo Wh he sffess cosas defed, he goverg eqaos of oo ca be wre sg Newo s Secod Law for each of he asses he syse: e ass acceerao forces acg o he ass && && && where: d & d Page -5
CEE49b Page -6 The for ass, geera for, where s he ber of he degrees of freedo: & & Ths s a se of saeos, ordary dfferea eqaos of he secod order Ths ca be wre arx for: [ ]{} [ ]{} {} & & where [ ] s he dagoa ass arx: [ ] The dspacee vecor s a co arx: { } [ ] T ad he syerca sffess arx [] s: [ ] As he sge degree of freedo case, he parcar soo s: s s && or, arx oao: { } { } { } { } s s &&
CEE49b I whch { } s he co vecor of apdes, whch are depede of e Sbsg eqao o eqao, yeds: [ ] {} & s [ ]{ } s { } The co vecor of apdes s: 4 {} [ ] T Ths s a hoogeeos agebrac eqao for, where he freqecy, s ow Ths s caed he Egevae Probe The soo o he Egevae Probe zes he basc properes of hoogeeos agebrac eqaos whch py ha he roos ows are orva e o eqa o zero oy f he deera of he coeffces vashes [ ] [ ] { } { } The roos of hs eqao are o-zero oy f he deera s zero e: [ ] [ ] f λ / ad we pre-py by λ [ ], he: λ[ ] [ ] [ ] [ ] { } { } λ[] I [ ] [ ] {} {} where [ I ] s he Idey Marx A o-rva soo oy exss f he deera s eqa o zero: λ [] I [ ] [ ] To fd he vaes of λ whch sasfy hs eqao ress a se of qe Egevaes or Nara Freqeces Afer he Egevaes have bee deered, hey are sbsed bac o he Hoogeeos Eqao vovg { } For each vae of λ reca ha λ / or ara freqecy, a copee se of Page -7
CEE49b Page -8 desoess dspacees are obaed, oe for each degree of freedo There are he ode shapes assocaed wh each ode of vbrao, caed Egevecors I or wo-degree of freedo fagpoe probe, he soo of hese eqaos ress a cosed-for soo for, as foows: Reca ha we ow have a x Codesed Sffess Marx, [ ], ad he eqao for he characersc deera: [ ] [ ] becoes: he deera s he a qadrac eqao : 4 4 he soo of whch s:, 4 ± Wh each freqecy, he apde raos, or ode shapes ca be cacaed fro: [ ] [ ] { } { }
CEE49b Ths ress wo eqaos ad wo ows: f we defe he reave dspacee as a a, he ad a a Boh of hese eqaos s yed he sae aswer, whch acs as a chec o he resg ode shapes Wh ore degrees-of-freedo ha wo, s desrabe o se a coper o sove he probe Page -9