425. Calculation of stresses in the coating of a vibrating beam

Transcription

1 45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. 45. Calulaton of stresses n the oatng of a vbratng beam M. Ragulsks,a, V. Kravčenken,b, K. Plkauskas,, R. Maskelunas,a, L. Zubavčus,b, P. Paškevčus,d Kaunas Unversty of ehnology, Studentų 50-, L-568, Kaunas, Lthuana e-mal: a mnvydas.ragulsks@ktu.lt b voleta.kravenkene@ktu.lt kestuts.plkauskas@ktu.lt Phone: ; Fax: Vlnus Gedmnas ehnal Unversty, J. Basanavčaus 8, L-05, Vlnus, Lthuana e-mal: a rmas.maskelunas@me.vgtu.lt b leonas.zubavus@me.vgtu.lt (Reeved 07 January 009; aepted 0 Marh 009) Abstrat. he model for the analyss of stresses n the oatng of a straght beam performng transverse vbratons s presented. he developed numeral proedure s based on the tehnque of onjugate approxmaton wth adapted for ths spef problem. Relatve error norms are alulated and ther relatonshp wth the requred s determned. Keywords: beam vbratons, egenmode, onjugate approxmaton, onjugate, fnte elements. Introduton he numeral alulaton and analyss of stresses n photo-elast oatngs s mportant n hybrd expermental numeral proedures used n expermental mehans [, ]. One of the smplest and mportant models n ths feld s the analyss of transverse vbratons of a straght beam. It s assumed that the oatng s thn and has no effet on the vbratons of the beam. he model for the analyss of stresses n the oatng of a straght beam performng transverse vbratons s presented. It s based on the model for the analyss of beam bendng desrbed n []. he developed numeral proedure s based on the tehnque of onjugate approxmaton [4, 5] wth [6] adapted for ths spef problem. Relatve error norms [7, 8] are alulated and ther relatonshp wth the requred s determned. he alulaton of the stress feld s mportant n hybrd expermental numeral proedures, whh were analyzed n the prevous papers [9, 0]. One dmensonal model whh s analyzed n ths paper enables to represent the results graphally and thus provdes better understandng of the behavor of approxmaton as well as neessty of of the stress feld. Numeral proedure Further x, y and z denote the axes of the orthogonal Cartesan system of oordnates. he beam bendng element has two nodal degrees of freedom: the dsplaement w n the dreton of the z axs and the rotaton Θ y about the y axs. he dsplaement u n the dreton of the x axs s expressed as u=zθ y. VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 009 MARCH, VOLUME, ISSUE, ISSN 9-876

2 45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. he mass matrx has the form: ρh 0 = dx, 0 ρ [ M] [ N] h [ N] where ρ s the densty of the materal of the beam, h s the thkness of the beam and: N [ ] =, N () N () where N are the shape funtons of the fnte element. he stffness matrx has the form: where: [ K] E h [ B] [ B] + ν = dx, E [ ] [ ] + B h ( ) B + ν. dn dn = N [ ] = 0..., () B (4) [ ]..., B (5) where E s the modulus of elastty of the beam, ν s the Posson s rato of the beam. he -th egenfrequeny ω and the orrespondng egenmode {δ } are determned from: ω δ (6) where: ([ K] [ M] ){ } = { 0}. he nodal values of the stress n the oatng are determned from: [ Nˆ ] [ Nˆ ] [ Bˆ ] [ Bˆ ] dx{ δ } = [ Nˆ ] + σ dx, [ ˆ] [ N ], [ ˆ dn ] =, λ (7) N = (8) B (9) where λ s the parameter, {δ } s the vetor of nodal values of the stress n the oatng, σ s the stress n the oatng determned from: σ h ν [ B]{ }, = (0) δ where E s the modulus of elastty of the oatng, ν s the Posson s rato of the oatng, {δ} s the vetor of generalzed dsplaements of the analyzed egenmode. he relatve error norms for the -th fnte element are alulated as: VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 009 MARCH, VOLUME, ISSUE, ISSN 9-876

3 45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. = h ε ν ε dx e ν [ B]{ δ} e ψ () dx, where the ntegrals are over the analyzed fnte element and the values of ε are alulated from: σ =, ν ε () where the stress n the oatng s alulated by usng the shape funtons of the fnte element from the vetor {δ } obtaned by usng the proedure of onjugate approxmaton (λ=0). Results of analyss At both ends of the beam both generalzed dsplaements are assumed equal to zero. he stress feld for the frst egenmode obtaned by usng the proedure of onjugate approxmaton s presented n Fg. a, the hstogram of the relatve error norms s presented n Fg. b, and the stress feld obtaned by usng the proedure of onjugate s presented n Fg.. he orrespondng results for the seond egenmode are presented n Fg.,, for the sxth egenmode - n Fg. 6. a) b) ) Fg.. he frst egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate a) b) ) Fg.. he seond egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 009 MARCH, VOLUME, ISSUE, ISSN 9-876

4 45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. a) b) ) Fg.. he thrd egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate a) b) ) Fg. 4. he fourth egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate a) b) ) Fg. 5. he ffth egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate a) b) ) Fg. 6. he sxth egenmode: a) stress feld obtaned by usng the proedure of onjugate approxmaton, b) the hstogram of the relatve error norms, ) stress feld obtaned by usng the proedure of onjugate 4 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 009 MARCH, VOLUME, ISSUE, ISSN 9-876

5 45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. From the presented results the advantage of onjugate over the proedure of onjugate approxmaton and the relatonshp of the requred wth the relatve error norms are evdent. hs enables to propose varous algorthms for the adaptve ontrol of the parameter. Conlusons he model for the analyss of stresses n the oatng of a straght beam performng transverse vbratons s presented. he developed numeral proedure s based on the tehnque of onjugate approxmaton wth adapted for ths spef problem. Relatve error norms are alulated and ther relatonshp wth the requred s determned. From the presented results the advantage of onjugate and the relatonshp of the requred wth the relatve error norms are evdent. hs enables to propose varous algorthms for the adaptve ontrol of the parameter. Referenes [] Holsten A., Salbut L., Kujawnska M., Juptner W. Hybrd expermental numeral onept of resdual stress analyss n laser weldments. Expermental Mehans, 00, 4(4), [] Kobayash A. S. Handbook on Expermental Mehans. SEM, 99. [] Bathe K. J. Fnte Element Proedures n Engneerng Analyss. New Jersey: Prente-Hall, 98. [4] Segerlnd L. J. Appled Fnte Element Analyss. Mosow: Mr, 979. [5] Gallagher R. H. Fnte Element Analyss Fundamentals. Mosow: Mr, 984. [6] Ragulsks M., Kravčenken V. Adaptve onjugate of dsontnuous felds. Leture Notes n Computer Sene, 005, , [7] Barauskas R. Bass of the Fnte Element Method. Kaunas: ehnologja, 998. [8] Barauskas R., Belevčus R., Kačanauskas R. Bass of the Fnte Element Method. Vlnus: ehnka, 004. [9] Saunoren L., Ragulsks M., Maskelūnas R., Zubavčus L. Analyss of bendng vbratons of a plate usng photo-elast oatngs. Journal of Vbroengneerng, 005, 7(), -6. [0] Ragulsken J., Maulevčus J., Ragulsks K., Zubavčus L. Valdaton of vbraton analyss usng photo-elast oatngs. Journal of Vbroengneerng, 007, 9(), -5. VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 009 MARCH, VOLUME, ISSUE, ISSN