Damped Vibration of a Non-prismatic Beam with a Rotational Spring
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1 Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa, Polad sochacki@imipkm.pcz.pl Marta BOLD stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa, Polad bold@imipkm.pcz.pl Abstract this paper a problem pertaiig to the damped lateral vibratios of a beam with differet boudary coditios ad with a rotatioal sprig is formulated ad solved. the adopted model the vibratio eergy dissipatio derives from the iteral dampig of the viscoelastic material (Kelvi Voigt rheological model) of the beam ad from the resistace motio i the supports. The rotatioal sprig ca be mouted at ay chose positio alog the beam legth. The ifluece of step chages i the cross-sectio of the beam o its damped lateral vibratios is also ivestigated i the paper. The damped vibratio frequecy ad the vibratio amplitude decay level are calculated. Chages i the eigevalues of the beam vibratios alog with the chages i the dampig ratio ad the chage i the model geometry observed o it are also preseted. The cosidered beam was treated as Euler- Beroulli beam. Keywords: Vibratio dampig, o-prismatic beam, rotatioal sprig.. troductio The trasverse vibratio of prismatic ad o-prismatic beams with additioal discrete elemets has bee ivestigated i a umber of studies. Study [] presets the trasverse vibratio of a beam with a stepped cross-sectio together with the pheomeo of damped vibratio i the body where the system is preset. The problem of the vibratio ad dyamic stability of beams with differet boudary coditios with additioal discrete elemets was preseted i study []. Study [3] cocered the modal aalysis of a semi-ifiite Euler-Beroulli beam with discrete elemets i the form of a rotatioal ad a traslatioal sprig. vestigatios cocerig damped vibratio were discussed i [-7]. Study [] discussed the effect of small iteral ad exteral dampig o the stability of o-coservative beam systems. The authors of study [5] demostrated the effect of iteral dampig o the vibratios of a supported beam with a mass attached to the free ed of the beam. Study [6] examied the vibratio of a axially-loaded Timosheko beam with local iteral dampig. The effect of costructioal dampig of the fixatios o free vibratio of the Beroulli-Euler beam was preseted i study [7]. This study formulates ad solves the problems of trasverse damped vibratio i a C- P (clamped-pied) beam with a stepped cross-sectio ad with a rotatioal sprig. Dissipatio of the vibratio eergy occurs as a result of the simultaeous iteral dampig
2 5 of the viscoelastic material of the beam ad structural dampig i the support. The costructioal dampig was modelled usig a rotatioal viscous damper. The study aalyses the simultaeous effect of the structural dampig ad iteral dampig, the sprig rigidity ad its locatio ad the effect of the locatio of the stepped cross-sectio of the beam o the properties of the cosidered system. The results obtaied i the study are preseted as D figures ad spatial presetatios.. Mathematical model A scheme of the cosidered C-P beam is preseted i Fig.. Figure. Model of the C-P beam with step chages i the cross-sectio with a rotatioal sprig C S ad rotatioal viscous damper C R Viscoelastic material was characterized by the Youg's modulus E ad the viscosity coefficiet E of the beam material. The coefficiet of costructioal dampig i the rotatioal viscous damper was deoted as C R. The vibratio equatio for the two parts of a beam is kow ad has the followig form: + P W EJ + E J x W + ρ A x W (x, the lateral displacemet of beam, A the cross-sectio area of the beam, J the momet of iertia for the beam sectio, ρ the desity of the beam material, P logitudial forces i beam, =, x space coordiate, t time, W + x t 5 W = 0 t ()
3 Vibratios i Physical Systems Vol.6 (0) 53 Solutios to equatios () take the form: iω t ω the complex eigevalue of the system, i = Substitutio of () ito () leads to: Boudary coditios: w (0) = 0, w V W = w ( x) e () ( x) + β w ( x) γ w ( x) = 0 (3) ω ω ρ A P γ =, β ( E + ie ) J = ( E + ie ω ) J () w ( l ) = w (0), (E + ie ω )J w ( l ) = w (0), w (E + ie ω )J w (0) = 0, (l ) + (E + ie ω )J ( 0 ) = 0, w ( l ) = 0, (E + ie ω )J w ( 0 ) + C w ( l ) = (E + ie ω )J S w w (l ) = C R i ω w (l ), w ( 0) The solutio to equatios (3) is expressed i the form of fuctios: (5) λ x λ x iλ x iλ x = D e + De + D3e + De w ( x) (6) β β λ = + + γ β, β λ = + + γ By substitutig (6) ito (5) a homogeeous system of equatios was obtaied with respect to ukow costats D k, ad ca be writte i the matrix form as: ( ) [ ] A = a pq [ ]( ) D = 0 (7) A ω (8) ω, ( p, q =,..8), [ ] T D =, k =, (9) The system has a otrivial solutio whe the matrix determiat of coefficiets is equal to zero with costats D k. ( ω ) = 0 D k det A (0) Fidig the complex eigevalues of matrix A(ω ) leads to the determiatio of damped vibratio frequecy Re(ω ) ad the vibratio amplitude decay level m(ω ) of the cosidered system.
4 5 3. Numerical calculatio results Computatios were carried out assumig the followig dimesioless quatities: E η =, ae a ρ A = LC, E J C L c =, ( E + )J S C ie ω J J =, J CR µ =, () L ρ C l L C A (E + ieω )J L =, P p = () P C P C the critical load of the tested beam with a costat cross-sectio. The results of the calculatios are preseted i Figs. to 6. vestigatios were carried out for differet ratios of the momets of iertia for the two parts of the beam (J=0.5, J=5) ad for a beam with a costat cross-sectio (J=). The system was loaded with logitudial force P (p=0.05). The depedecy of the eigevalues (real parts Re(ω ) ad imagiary parts m(ω )) o the coefficiets of costructioal dampig µ, sprig rigidity c ad locatio of the chage i the beam cross-sectio l was also determied. Figure. The depedecy of real parts (Re(ω )) ad imagiary parts (m(ω )) of the first beam eigevalue o the coefficiet l at η=0.00, µ=0.3, c=0 Figure 3. The depedecy of real parts (Re(ω )) ad imagiary parts (m(ω )) of the first beam eigevalue o the sprig rigidity coefficiet c at η=0.00, µ=0.3 ad l=0.
5 Vibratios i Physical Systems Vol.6 (0) 55 Figures 5 ad 6 preset collective diagrams of the depedecy of eigevalues (Re(ω ) ad m(ω )) i the studied system o the chage i the rigidity of elastic support c ad costructioal dampig µ. The calculatios were carried out for selected values of iteral dampig ad for a cetral locatio of the rotatioal sprig ad two values of the relatio of the momets of iertia (J=5 ad J=0.5). The results are preseted as spatial diagrams. Figure. The depedecy of real parts (Re(ω )) ad imagiary parts (m(ω )) of the first beam o structural dampig µ at η=0.00, c=0 ad l=0. Figure 5. The depedecy of real parts (Re(ω )) ad imagiary parts (m(ω )) of the first eigevalue of the beam o the coefficiet of structural dampig µ ad sprig rigidity coefficiet c for l=0.5 ad J=0.5, η=0.00 Figure 6. The depedecy of real parts (Re(ω )) ad imagiary parts (m(ω )) of the first eigevalue of the beam o the coefficiet of structural dampig µ ad sprig rigidity coefficiet c for l=0.5 ad J=5, η=0.00
6 56. Coclusios The damped frequecies of system Re(ω ) ad the degree of amplitude decay m(ω ) i the system deped o the locatio of the rotatioal sprig alog the beam. No uiform tedecy for chages was observed i the case studied (Fig. ). mproved sprig rigidity causes a costat icrease i the damped frequecies of the first eigevalue of the system (for selected values of coefficiets η, µ ad l). The degree of amplitude decay i this case depeds o the ratio of rigidity J for the two beam parts. For the cetral locatio of the chage i the cross-sectio (l=0.5), a icrease i c causes a decrease i the coefficiet of the amplitude decay for J=0.5, ad a icrease for J=5 (Fig. 5 ad 6). The costructioal dampig of the fixatio poits with selected values of sprig rigidity causes much more substatial chages i the eigevalues of the system tha i the reverse case (the chage i coefficiet c for selected value µ). The results preseted i the study help determie the geometric parameters ad values of the coefficiets that characterize the dampig ad elasticity of the system for which the maximum degree of amplitude decay is maitaied. Ackowledgmets This research was supported i 0 by the Miistry of Sciece ad Higher Educatio, Warsaw, Polad. Refereces M.. Friswell, A. W. Lees, The modes of o-homogeeous damped beams, Joural of Soud ad Vibratio () (00) W. Sochacki, Dyamic stability of discrete-cotiuous mechaical systems as workig machie models, Moographic series 7, Czestochowa Uiversity of Techology, Czestochowa 008. A. Pau, F. Vestroi, Modal Aalysis of a Beam with Radiatio Dampig: Numerical ad Experimetal Results, Joural of Vibratio ad Cotrol 3(8) (007), O. N. Kirrllov, A. O. Seyrai, The effect of small iteral ad exteral dampig o the stability of distributed o-coservative systems, Joural of Applied Mathematics ad Mechaics 69 (005), M. Gürgöze, A. N. Doğruoğlu, S. Aere, O the eigecharacteristics of a catilevered visco-elastic beam carryig a tip mass ad its represetatio by a sprigdamper-mass system, Joural of Soud ad Vibratio 30 (007), 0-6. Wei-Re Che, Bedig vibratio of axially loaded Timosheko beams with locally distributed Kelvi Voigt dampig, Joural of Soud ad Vibratio 330 (0), G. Oliveto, A. Satii, E. Tripodi, Complex modal aalysis of flexural vibratig beam with viscous ed coditios, Joural of Soud ad Vibratio 00 (997),
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