Loose Waves in Viscoelastic Cylindrical Wave Guide with Radial Crack

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1 Applied Mathematics, 014, 5, Published Online Decembe 014 in Scies. Loose Waves in Viscoelastic Cylindical Wave Guide with adial Cack Ismail Ibagimovich Safaov, Maqsud Shaipovich Akhmedov, Zafa Ihteovich Boltaev Bukhaa Technological-Institute of Engineeing, Bukhaa, epublic of Ubekistan eceived 6 Octobe 014 evised Novembe 014 accepted 16 Novembe 014 Copyight 014 by authos and Scientific eseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY. Abstact The main featues ae the length of the waveguide in one diection, as well as limitations and localiation of the wave beam in othe aeas. Thee is descibed the technique of the solution of tasks on distibution of waves in an infinite cylindical waveguide with a adial cack. Also numeical esults ae given in the aticle. Viscous popeties of the mateial ae taken into account by means of an integal opeato Voltaie. eseach is conducted in the famewok of the spatial theoy of visco elastic. The technique is based on the sepaation of spatial vaiables and fomulates the bounday eigenvalue poblem that can be solved by the method of othogonal sweep Godunov. In the given pape we obtain numeic values of the phase velocity depending on of wave numbes. The obtained numeical esults ae compaed with the known data. This wok is continuation of aticle [1]. Statement of the poblem and methodology of patial solutions ae descibed in [1]. In this wok, we pesent a complete statement of the poblem, methods of solution and discuss the numeical esults. Keywods The Wave Guide, Wave, Cylinde, Cack, Integal Opeato, Diffeential Equations, elaxation Kenel Othogonal Sweep, Appoximation, Patial Deivatives, The Phase Velocity, Fequency, Damping Facto 1. Intoduction Popagation of waves in a cylinde was the subject in numeous theoetical and expeimental investigations. Wave popagation in the cylinde was investigated by Pochhamme L. [] and Chee Ch. [3]. Fo an elastic laye (two-dimensional poblem simila esults wee obtained by ayleigh [4]. The fist numeical esults elating to some of the chaacteistics of nomal waves in the laye wee given in Lamb H. [5]. The poblem of wave popagation in a solid cylinde is investigated in Hinchenko and Myaleshka V.V. [6]. Many constuction and engi- How to cite this pape: Safaov, I.I., Akhmedov, M.Sh. and Boltaev, Z.I. (014 Loose Waves in Viscoelastic Cylindical Wave Guide with adial Cack. Applied Mathematics, 5,

2 I. I. Safaov et al. neeing design wok in dynamic conditions consist of defomable bodies with viscoelastic (damping popeties. Damping capacity of the mateial plays an impotant ole in the dynamic behavio of the stuctue. This leads to a significant weakening of the natual oscillation and educes the oscillation amplitude. The damping capacity of the mateial plays a huge ole in the dynamic behavio of the stuctue. The wok deals with hamonic waves in a spatial cylinde of the adial cacked consideing the damping capacity of the mateial.. Statement of the Poblem This wok examined the distibution of fee waves in an infinite cylinde with adial cacks with oute and inne adii. Viscoelastic cylinde with a adial cack is a limiting case of a wedge with an angle of 360. The poblem of popagation of hamonic waves in the infinite elastic cylinde with adial cack is put in cylindical coodinates of, and [7]: u σ σ σ 1 σ σ ρ = dt d u 1 σ σ σ σ ρ = (1 t u σ σ σ 1 σ ρ = t Hee whee u u 1 u u ε = ε = ε = u u u 1 u u 1 u 1 u ε = + ε = + ε = + u 1 u u u u 1 u u u σ = σ = ε = + u u σ = ε = + u 1 u u u 1 u u σ = u 1 u σ = + u 1 u u u u σ = , t f ( t = 0 f ( t ( t τ f ( τ d τ, 0 (4 t f ( t = 0 f ( t ( t τ f ( τ d τ 0 f ( t a function ρ density mateials, ( t τ and ( t τ the coe elaxation, 0, 0 the instantaneous modulus of elasticity of a viscoelastic medium, u ( u, u, u is the vecto displacement which depends on σ, σ, σ, σ, σ, σ the components of the stess tenso ε, ε, ε, ε, ε, ε the components of the stain tenso. Equation (4 afte the application of the method of feeing [8] takes the following fom: ( (3 3519

3 I. I. Safaov et al. C S C S ( = Γ ( Γ ( ( ( = Γ ( Γ ( ( f t 1 ω i ω f t, f t m 1 ω i ω f t, C S C whee Γ ( ω = ( τ cosωτdτ, Γ ( ω = ( τ sinωτdτ, Γ ( ω = ( τ cosωτdτ, S 0 ( ( Γ ω = τ sinωτdτ, the cosine and sine Fouie tansfoms, espectively ω the eal pat of the complex fequency ( ω = ω + iωi ρ density ( t and ( t espectively elaxation kenel mateial. elations (1, (, (3 ae identical to the system of six diffeential equations afte algebaic manipulations complex coefficients ae solved fo the fist deivative with especting to the adial coodinate. u 1 1 u u u = σ + + K K u u σ φ u = u u = σ σ u A 1 σ σ = ρ t σ u 1 σ = ρ σ A B t σ u u u 1 = ρ σ σ B t u 1 u whee A u 1 u = + u B = +, The bounday conditions ae specified as: ν K =. 1 ν = 0 0, : σ = σ = σ = 0 (6 = 0, π : = 0 σ = σ = 0 (7 u Condition (3 at = 0 in the physical plane can be intepeted as the esult of a limiting tansition fom of a hollow cylinde with a fee inne suface to the solid, the inne adius tends to eo. In the case of hamonic waves taveling along the axis, the solution of (5, (6 and (7 allows the sepaation of vaiables: iк( ct u = w ( cos e iк( ct u = ϑ ( sin e iк( ct u = u( cos e (8 iк( ct σ = σ ( cos e iк( ct σ = τ ( sin e iк( ct σ = τ ( cos e,, ϑ,, σ, τ, τ oscillation amplitude that ae a function of adial coodinate к Complex wave numbe c complex phase velocity ω the complex fequency. whee w ( ( u ( ( ( ( (5 350

4 I. I. Safaov et al. To claify the physical meaning of the given case: 1 к = α ω = ω + iωi ( C = C + ici then the solution (5 has the fom sinusoid on, the amplitude of which decays ove time к = α + iαi ω = ω ( C = C then the solution (5 has the fom sinusoid in t, whose amplitude attenuates In view of (8, the poblem (6, (7 ae tansfomed into a spectal bounday value poblem fo a system of odinay diffeential equations with complex the coefficient: σ ϑ w w = ku + + K K τ ϑ w v = + + τ u = + kw a τ σ = ωρw + kτ τ ( σ + a τ = ω ρϑ + kb τ b τ = ωρu + k ( σ + ( ku w, (9 whee (... d v+ w u =, a = w b = kϑ. d On the bounday conditions at = 0 0, =, σ = τ = τ = 0. (10 Thus we have obtained the spectal poblem (9 and (10 descibing the popagation of hamonic waves in infinite elastic cylinde with a adial cacks. Note, that the choice of bounday conditions on the edges of the slit in the fom (4 to detemine fist of all the possibility of sepaation of vaiables and the coodinates, which geatly simplifies the solution of the oiginal poblem. Sepaation may also be displaced in the event of the following bounday conditions: = 0, π σ = 0, u = u = 0 (11 Indeed, pefoming the change of vaiables so as to satisfy the condition (11 iк( ct u = w ( cos e iк( ct uφ = ϑ ( sin e iк( ct u = u ( cos e iк( ct σ = σ ( cos e iк( ct σ = τ ( sin e iк( ct σ = τ ( cos e, Obtain the spectal bounday value poblem 351

5 I. I. Safaov et al. with bounday conditions: σ ϑ w w = ku + K K τ ϑ w ϑ = + τ u = + kw ϑ w τ σ = ρω w + + w + k τ τ 1 ϑ w π τ = ρω ϑ σ + + w k k ϑ τ u τ = ρω u + k ϑ k ( σ ( ku w + + (1 = 0 0,, σ = τ = τ = 0 (13 Not difficult to see that the poblem (1, (13 is educed to the poblem (9, (10 by eplacing. w = w, u = u, u = u, σ = σ, τ = τ, τ = τ Thus, we obtained the spectal poblem (10 descibing of the popagation of hamonic wave in an infinite cylinde with a adial cack. A bounday value poblem fo a system equation in patial deivatives (10 can be kept to bounday value poblem fo a system of odinay diffeential equations using method of lines, which will be used in solving a softwae unit othogonal sweep method. Accoding to the method of a ectangula domain of definition of the basic unknowns was coveed by lines paallel to the -axis and evenly spaced. The solution is sought only on these line sand the deivative in the diection of φ, is eplaced appoximations finite diffeences. Useful appoximate fomula of second ode fo fist and the second deivative have the fom: yi+ 1 yi 1 3yi + 4yi+ 1 yi+ 3yi 4yi 1+ yi yi, κ (14 whee i vaies fom 0 to N 1 ( i 0, N 1 y y + y i+ 1 i i 1 yi, + = +, y i the pojection of the unknown function on the line with the numbe i Δ step beakes coodinate φ. As a esult, the main vecto of the unknown simple ove all dimensions 6N can be witten as [9]: ({ } { } { } { } { } { } T i υi i σi τi τi Y = w,, u,,, i = 1, N (16 By cental diffeences (14 and (15 ae used fo intenal lines (1 < i < N, the diffeence left and ight allow taking into account the bounday conditions on φ. In the fist case, the deivative with espective to φ in the ight sides of Equations (9 can be expessed by the fomulas: 1 < i < N ( +, ( + ( + ( + ( w = w w u = u u i, i 1 i 1 i, i 1 i 1 ( υ = υ υ, τ = τ τ, i, i 1 i 1 i, i 1 i 1 b σ, ( 1 1 ( 1 1 i = a σi+ σi + vi+ υi + vi + w i, + ckui, 0 The bounday conditions ae = taken into account in the equations coesponding to i = 1 lines. Fo (15 (17 35

6 I. I. Safaov et al. basic unknowns, non-bounday conditions, w i, v i, u i ae used ight diffeence: ( υ ( υ υ υ ( w = 3w + 4w w, = 3 + 4, u = 3u + 4u u (18 i, 1 3 i, 1 3 i, 1 3 Similaly, the deivatives ae pesented fo the fowad with numbe i = N, taking into account the bounday 0 conditions at =. The only diffeence is the eplacement of the left ight finite diffeences: i = N: ( υ ( υ (, ( 3, w = 3w 4w + w, = 3, i, N N 1 N i, N u = u u u = u i, i+ 1 i 1 i, N ( τ = τ, τ = τ τ, (19, ( N 1, ( i+ 1 ( i 1 i i b σ σi, = aσn 1+ ( υn υn + wn 1 + ckun 1 = If anti of symmety the conditions of the tansvese oscillations of the cylinde is used, the numbe of lines possible to educe by half. Thus, the initial spectal poblem (9 by means of sampling coodinates φ educed to the poblem (14-( The Numeical esults The solution of (9 is satisfied by othogonal sweep Machuk [9]. Solution of the poblem on the detemination of the velocity of wave popagation and wave fom is caied out by method of Mulle [10], without isolation of the complex paamete C (complex phase velocity, o ω the complex fequency. On each step the bounday-value poblem (9 is solved by method of othogonal po-ace. Phase velocity (C and own numbes (ω ae defined fom an equality condition to eo deteminant of system and ae calculated by the Gauss method. Dimensionless paametes in the poblem ae chosen so that the eal and imaginay pats of the complex shea ate С s the density ρ and oute adius have the single value, A = 0.00, α = 0.05, β = Figue 1 shows of the dispesion cuves of the fist two oscillation modes infinite elastic cylinde with the N 1 Figue 1. Change in the eal (С and imaginay (С I pats of the phase velocity fom. wave numbes. 353

7 I. I. Safaov et al. Figue. Change in the eal and imaginay pats mode shapes V and V I by к. adial factue (cuve 1 and. Fo compaison, the illustation shows dependence of the wave numbe of the phase velocity of the flexual vibation modes of the solid cylinde (cuve 3 without a gap. Pohgomeom and Cee using special functions aleady found solution of the poblem ealie. We note the chaacteistics of the cuve 3 in eo phase velocity is eo and infinity appoaches the velocity of the ayleigh wave fo a half. In the case of a cylinde with the adial cacked pats fist the eal fashion has a cutoff fequency, and the phase velocity tends to infinity. At lage wave numbes limit the phase velocity of this mode is also equal to the velocity of the ayleigh wave. At the cutoff fequency of the axial displacement to eo vibation of the cylinde is held in plane stain condition. In this pape, this solution was used fo testing tasks. In the second mode at the cutoff fequency obseved only axial movement, cicula and adial displacements ae eo. The evolution of foms of movement on the fist and second modes, depending on the wave numbe is shown in Figue. The cuves ae numbeed in inceasing ode of k. Note the stong dependence on the wave numbe of foms. With the gowth of the wave, numbe fo the fist mode is localied oscillations nea the oute suface of the cylinde. It is typical that the second mode, which is on the small wave numbes, is a fom of pedominantly axial vibation, with gowth to gadually tun into a fom of pedominantly adial oscillations. Accounting fo the viscoelastic popeties of the mateial allows educing the amplitude of the of wave popagation by 10% - 15%, as well as allows us to estimate damping ability of the system as a whole. efeences [1] Safaov, I.I., Akhmedov, M.Sh., Nuiddinov, B.Z. and Shaipov, D.Sh. (014 Waves in Viscoelastic Cylinde with the adial Cack. Magaine Young scientist, 4, [] Pochhamme, L. (1876 Übe die Fotpflanung segechwindigkeiten schwingungen in einem unbegawten isotopen keiscylinde. J.eine und angew. Math, 81, [3] Ci, C. (1886 Longitudinal Vibations of a Cocabla Ba. Quat. J. Pue and Appl. Math, 1, [4] ayleigh, J.W. ( On Waves Popagated along the Plane Suface of an Classic Solid. Poceedings of the London Mathematical Society, 17, [5] Lamb, H. (1917 On Waves in an Elastic Plate. Poceedings of the oyal Society A, 93, [6] Hinchenko, V.T. and Meleshko, V.V. (1981 Hamonic Oscillations and Waves in Elastic Bodies. Naukova Dumka, Kiev, 83 p. [7] Safaov, I.I., Teshaev, M.K.H. and Boltaev, Z.I. (01 Wave Pocesses in a Mechanical Waveguide. LAP LAMBET Academic Publishing (Gemany, 17 p. [8] Kavchuk, A.S., Majbooda, V.P. and Uhumaev, Y.S. (1985 Mechanics of Polymeic and Composite Mateials: Expeimental and Numeical Methods. Publishing Science, Moscow, 304 p. [9] Machuk, G.I. (1977 Methods of Computational Mathematics. Publishing Science, Moscow, 456 p. [10] Booov, M.B., Safaov, I.I. and Shokin, Y.I. (1996 Numeical Simulation of Oscillations Dissipative Homogeneous and Inhomogeneous Mechanical Systems. Publishe Sibeian Banch of the Academy of Sciences, Novosibisk, 188 p. 354

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