Longitudinal Waves in a Rotating Solid Cylinder Immersed in an Inviscid Fluid Using Chebyshev Polynomial

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1 Longituinal Waves in a Rotating Soli Cyliner Immerse in an Invisci Flui Using Chebyshev Polynomial R. Selvamani *1, P.Ponnusamy 1 Department o Mathematics, Karunya University, Coimbatore, TamilNau, Inia Department o Mathematics, Govt Arts College, Coimbatore, TamilNau, Inia *1 selvam179@gmail.com Abstract-The longituinal waves in a homogeneous transversely isotropic rotating soli cyliner immerse in an in visci lui is stuie using the linearie, three-imensional theory o elasticity. The equations o motion o soli an lui are respectively ormulate using the constitutive equations o a transversely isotropic cyliner an the constitutive equations o an in visci lui. The solution o the requency equations are obtaine by Chebyshev polynomials series using the geometric bounary conitions. The compute non-imensional requencies are presente in the orm o ispersion curves or the material Zinc. Keywors- Soli Cyliner Loae with Flui; Free Vibration o Transversely Isotropic Cyliner; Wave Propagation in a Rotating Cyliners; Chebyshev Polynomials I. INTRODUCTION Soli cyliner immerse in lui is the practical element o many types o engineering structure such as marine crat an airplanes in which the machine inuce vibrations oten occurs. It is thereore very important that the static an ynamic behavior o plates when subjecte to ierent loaing conitions be clearly unerstoo so that they may be saely use in these inustrial applications. It is well known that the natural requencies o structures in contact with lui are ierent rom those in vacuo. The longituinal moes oten use to evaluate the material properties o thin metal wires, reinorcement ilament in ultrasonic transucers an resonators. Applying Chebyshev polynomial series as the amissible unction or each isplacement component has istinct avantages like rapi convergence an better numerical stability in computation than other algebraic polynomial series. The most general orm o harmonic waves in a hollow cyliner o circular cross section o ininite length has been analye by Gais [1]. Mirsky [] investigate the wave propagation in transversely isotropic circular cyliners o ininite length an presente the requency equation in Part I an numerical results in Part II. A metho, or solving wave propagation in arbitrary cross-sectionl cyliners an plates an to in out the phase velocities in ierent moes o vibrations namely longituinal, torsional an lexural, by constructing requency equations was evise by Nagaya [3-5]. He ormulate the Fourier expansion collocation metho or this purpose. Following Nagaya, Paul an Venkatesan [6] stuie the wave propagation in an ininite pieoelectric soli cyliner o arbitrary cross section using Fourier expansion collocation metho. The longituinal waves inhomogeneous anisotropic cylinrical bars immerse in a lui are stuie by Dayal [7]. Guie waves in a transversely isotropic cyliner immerse in a lui are analye by Ahma [8]. Following Ahma, Nagay [9] have stuie the longituinal guie wave propagation in a transversely isotropic ro immerse in lui, later, Nagy with Nayeh [10] iscusse the viscosity-inuce attenuation o longituinal guie waves in lui-loae ros. Easwaran an Munjal [11] reporte a note on the eect o wall compliance on lowest-orer moe propagation in lui-ille/submerge impeance tubes. Sinha et. al. [1] have iscusse the axisymmetric wave propagation in circular cylinrical shell immerse in lui, in two parts. In Part I, the theoretical analyses o the propagating moes are iscusse an in Part II, the axisymmetric moes excluing torsional moes are obtaine theoretically an experimentally an are compare. Berlinear an Solecki [13] have stuie the wave propagation in lui loae transversely isotropic cyliner. In that paper, Part I consists o the analytical ormulation o the requency equation o the couple system consisting o the cyliner with inner an outer lui an Part II gives the numerical results. Venkatesan an Ponnusamy [14] have obtaine the requency equation o the ree vibration o a soli cyliner o arbitrary cross section immerse in a lui using Fourier expansion collocation metho. Ponnusamy an Selvamani [15] iscusse the ispersion analysis o a generalie magneto thermo elastic cylinrical panel by introucing the thermal relaxation times. Later, Ponnusamy an Selvamani [16] investigate the eect o rotation in an axisymmetric vibration o a transversely isotropic soli bar immerse in an invisci lui using Bessel unction solutions. Glawell an Tahbilar [17] an Buchanan an Yii [18] use the inite element metho to stuy the 3D vibration o cyliners while Wang an Williams [19] compare their inite element result with - 4 -

2 the experimental ata. Singal an Williams [0] use simple algebraic polynomials as amissible unctions to stuy the 3D vibration o completely ree hollow cyliners an the theoretical results agree well with the experimental results. The same problem was also stuie by So an Leissa [1] using simple algebraic polynomials as amissible unctions, an high accurate results were given. Later, Zhou [] stuie the three imensional vibration analysis o soli an hollow circular cyliners using Chebyshev-Rit metho. In this paper, the longituinal waves in a homogeneous transversely isotropic rotating soli cyliner immerse in an invisci lui are stuie using the linearie, three-imensional theory o elasticity. Two isplacement potential unctions are introuce to uncouple the equations o motion an the solutions are obtaine by Chebyshev metho. The compute non-imensional requencies are presente in the orm o ispersion curves or the material Zinc. II. FORMULATION OF THE PROBLEM We consier a transversely isotropic soli cyliner o length L an the raius a immerse in invisci lui. The system is assume to be linear so that the linearie three-imensional stress equations o motion are use or both the cyliner an the lui. The system isplacements an stresses are eine by the cylinrical coorinates r, an. In cylinrical coorinates, the three-imensional stress equations o motion an strain-isplacement relations in the absence o boy orce are given as rr, r r r, r, rr (1a) ( ( u) u ) u 1 r 1, t, tt r r v (1b) 1 1 r, r,, r, tt 1 1 r, r r,, r r (1c) ( ( u) u ) w, t, tt The stress strain relation is given by c e c e c e (a) rr 11 rr 1 13 c e c e c e (b) 1 rr c e c e c e (c) 13 rr r c66er, c44e, r c44er () are the stress components, e rr, e, e, e r, e, e r are the strain components, c 11, c 1, c 13, c 33, c 44 an where rr,, r c66 c11 c1 are the ive inepenent elastic constants, is the mass ensity o the material. The isplacement equation o motion has the aitional terms with a time epenent centripetal acceleration ( u ) an u, t where, u ( u,0, w) is the isplacement vector an (0,,0) is a constant, the comma notation use in the subscript enotes the partial ierentiation with respect to the variables. The strain e ij are relate to the isplacements are given by e 1 rr u, r e r u v,, e w,, (3a) 1,,, e r u, w, r, e 1 u, r u, r r e v r v u (3b) in which u, an w are the isplacement components along raial, axial irections respectively. The comma in the subscripts enotes the partial ierentiation with respect to the variables. Substituting the Eqs. (3) an () in the Eq. (1), results in the ollowing three-imensional isplacement equations o motion:

3 1, rr, r c u r u r u r c c v , r c66 u, c44u, c44 c13 w, r 66 1, r t 1 r c c v u w, u, tt r c c u r c c u , r 66 11, c v r v r v r c v 1 66, rr, r 11, c v r c c w v 1 44, 44 13,, tt 1 1 rr r c44 c13 u, r c33 w, w u, t w, tt c w r w r w r c c u v 44,,, 44 13,, For longituinal wave, it is assume that the isplacement along the hoop irection, v is ero an Eqs. (4) reuces to 1 u w, t u, tt c u r u r u c u c c w 11, rr, r 44, 44 13, r (4a) (4b) (4c) (5a) 1 1 t tt c w r w r c c u c c u 44, rr, r 44 13, 44 13, r c w w u w 33,,, In case o real lui there is a riction between the particles which will maniest itsel in the orm o equal an opposite tangential or shearing orces on each particle at the common bounary. But in an invisci lui-soli interace, the perect-slip bounary conition allows iscontinuity in planar isplacement components. That is, the raial component o isplacement o the lui an soli must be equal an the longituinal components are iscontinuous at the interace. The above couple partial ierential equations are also subjecte to the ollowing non-imensional bounary conitions at the suraces r a. rr p r uu 0 (5b) (6) III. SOLUTION TO SOLID MEDIUM The Eq. (5) is couple partial ierential equations o the three isplacement components. This system o equations can be uncouple by eliminating two o the three isplacement components through two o the three equations, but this result in partial ierential equations o ourth orer. To uncouple the Eq. (5), we ollow Mirsky [] an assuming the solution o Eq. (5) as ollows: i, is the angular requency,, i( t) urt (,,) r, e (7a), r i( t) wrt (,,) ia W r, e (7b) where 1 r, W r, are the isplacement potentials an a is the geometrical parameter o the cyliner. By introucing the imensionless quantities such as, R a c 44, c11 c11 c44, r r a, / H 1 c13 c13 c44, v c44, c33 c33 c44,, c x r/ aan substituting Eq. (7) in Eq. (5), we obtain 1 where x x x c13 11 T t c a 44 / c 1c W 0 (8a) 1 W 0 (8b)

4 A non-trivial solution o the algebraic systems (7) exists only when the eterminant o Eqs. (7) is equal to ero c 1c W 1c13, 0 (9) where Eq. (9), on simpliication reuces to the ollowing ierential equation: A c11 4 A B C 0 (10), B 1c11, C (11) Solving the Eq.(10), the isplacement amplitue unctions r, an wr, are obtaine by the orthogonal series o Chebyshev polynomials multiplie by a geometry bounary unction or the longituinal moe as ollows: The constants r, Fu AP ij i rp i (1a) i1 j1, w ij ij i j W r F A P r P (1b) i1 j1 ij eine in the Eq. (1b) can be calculate rom the equation 1 13 ij c ia ia, i 1, Where the bounary unctions or the stress ree bounary conition is taken F 1 1 u Fu Fu Aij an B kl are the coeicients o the polynomial. Ps s 1,,3, 4,5 ; r, (13) orer o Chebyshev polynomial series, 1D s th Chebyshev polynomial which can be written in terms o cosine unction as ollows: =1 unity, an i, j are the is the Ps cos s1 arccos s 1,,3..., r, (14) IV. SOLUTION OF FLUID MEDIUM In cylinrical polar coorinates r, an, the acoustic pressure an raial isplacement equations o motion or an in visci lui are o the orm Berliner [13] an respectively where lui, an u, w is the isplacement vector. Substituting B, is the aiabatic bulk moulus, 1, r, p B u r ( u ) w (15) c u, tt, r (16) is the ensity, 1, r, c B is the acoustic phase velocity in the u r ( u ) w (17) u r, an, w (18) an seeking the solution o (15) in the orm i t (, r,,) t r e, (19)

5 the lui represents the oscillatory waves propagating away is given by A33 H ( ax) (0) n a B, Where in which, B B c 44, Hn is the Hankel unction o the irst kin. I ( a) 0, then the Hankel unction o irst kin is to be replace by K n, where K n is the moiie Bessel unction o the secon kin. The velocity an ensity relation between the cyliner an lui is eine by c 0 v c an 0, respectively. By substituting Eq.(18) in (15) along with (19) an (0), the acoustic pressure or the lui can be expresse as i Ta p A33 Hn ( ax) e (1) V. FREQUENCY EQUATIONS In this section we shall erive the requency equation or the three imensional vibration o the soli cyliner immerse in lui subjecte to perect slip bounary conitions at r a. Substituting the expressions in Eqs. (1)- (4) into Eqs. (6), we can get the requency equation or ree vibration as ollows: A c A 0 i, j 1,,3 ij 1 1 Fu 1 1 r Fu p r p p r p 11Fw p1 r p1 () A c Fu 1 r Fu p r p p r p 1Fw p1 r p A ( ax) nh ( ax) ( ax) H ( ax) 13 n n1 F E1 Fw 1 p r p1 u p r p1 r r F E Fw p r p u p r p E3 0 r r E nj ( ax) ( ax) J ( ax) 31 n 1 1 n1 1 E nj ( ax) ( ax) J ( ax) 3 n n1 E nh ( ax) ( ax) H ( ax) 33 n n1 VI. NUMERICAL RESULTS AND DISCUSSION The couple ree waves propagation in a homogenous transversely isotropic soli cyliner immerse in water is numerically solve or the Zinc material. The material properties o Zinc are given as ollows an or the purpose o numerical computation the liqui is taken as water. 11 c Nm 11, c Nm 11, c Nm, ensity kg m an or the lui the ensity 1000kg m c an phase velocity 11 Nm, c 11 Nm an 1 c ms

6 A comparison is mae or the non-imensional requencies in case o ree an immerse cyliner with respect to the velocity ratio or the symmetric an anti symmetric moes o the cyliner immerse in lui in Tables I an II. From these tables, it is clear that as the sequential number o the velocity ratio increases, the non-imensional requencies also increase or both the ree an immerse cyliner. When a soli meium such as the soli cyliner is surroune by lui meium, guie waves are transmitte across the interace. Thus bulk waves are excite in the embeing meium, raiating away rom the soli meium. TABLE 1 THE NON-DIMENSIONAL FREQUENCIES FOR FIRST THREE SYMMETRIC LONGITUDINAL MODES OF THE FREE AND IMMERSED CYLINDER WITH VELOCITY RATIO Velocity ratio(c0) Free cyliner Immerse cyliner S 1 S S 3 S 1 S S TABLE II THE NON-DIMENSIONAL FREQUENCIES FOR FIRST THREE ANTI SYMMETRIC LONGITUDINAL MODES OF THE FREE AND IMMERSED CYLINDER WITH VELOCITY RATIO Velocity ratio(c0) Free cyliner Immerse cyliner S 1 S S 3 S 1 S S VII. DISPERSION CURVES The result o the requency analysis or three longituinal moes o vibrations is plotte in the ollowing igures with respect to length to raius ratio o the cyliner. The notations LM1, LM an LM3 enotes the irst, secon an thir longituinal moes respectively. The variation o requencies with the length to raius ratio o the cyliner is iscusse in Fig.1 an Fig. or the ree soli cyliner with the rotational spee 0 an 0.5 or the irst three longituinal moes o vibration. In Fig.1 the requency increases monotonically in all the ranges length to raius ratio. Fig. 1 Variation o requency with L/a o ree soli cyliner with

7 For the case o rotational parameter 0.5 there is a small eviation on the requency pattern in Fig. ue to the extra riving orce exerte by the rotation o the cyliner. The energy transmission occurs only on the surace o the cyliner because the cyliner acts as the semi ininite meium. In general, rom the Figs.1 an it is observe that the non-imensional requency o the unamental moe is non-ispersive an increases rapily in the presence o rotational spee with increasing wave number. Fig. Variation o requency with L/a o ree soli cyliner with 0.5. In Fig. 3, the variation o requency with respect to length to raius ratio o the soli cyliner immerse in lui is presente or 0. The magnitue o the requency increases monotonically in the range 0 La 0.5 or irst three moes o the soli cyliner immerse in lui an become ispersive in the remaining range o L a ratios. Fig. 3 Variation o requency with L/a o immerse soli cyliner with 0 The variation o requency with respect to the length to raius ratio o the cyliner or irst three longituinal moes o the immerse soli cyliner with 0.5. is presente in Fig. 4, where the non imensional requency rises slow an steay in 0 La 0.4 an become ispersive ue to the combine eect o lui meium an rotational spee. From Figs. 3 an 4, it is clear that the eects o length to raius ratio o the soli cyliner are quite pertinent ue to the combine eect o rotation an amping eect o the lui meium. When the ratio o the ensities o the lui an elastic material is small (0.14), then the moe spectrum o lui loae cyliner is slightly ierent rom that o ree cyliner

8 Fig. 4 Variation o requency with L/a o immerse soli cyliner with 0.5 VIII. CONCLUSIONS In this paper, the longituinal vibration in a rotating homogeneous transversely isotropic soli cyliner immerse in an invisci lui is stuie using the linearie, three-imensional theory o elasticity. Two isplacement potential unctions are introuce to uncouple the equations o motion. In the present analysis, a set o Chebyshev polynomials multiplie by a bounary unction which satisies the geometric bounary conitions o the cyliner are taken as the trial unctions. The compute non-imensional requencies or ierent parameters are presente in the orm o ispersion curves or the material Zinc. From the tabulate values, it is clear that the eect o the lui orce on the structure is represente as ae mass, which lowers the natural requency o the structure rom that which woul be measure in a vacuum. This ecrease in the natural requency o the lui-structure system is cause by increasing the kinetic energy o the couple system without a corresponing increase in strain energy. REFERENCES [1] D.C.Gais, Three-imensional investigation o the propagation o waves in hollow circular cyliners. I.Analytical Formulation;II Numerical Results, J. Acoust. Soc. Am. 31, , [] I.Mirsky, Wave propagation in transversely isotropic circular cyliners. J. Acoust. Soc. Am. 36, 41-51, [3] K.Nagaya, Wave propagation in an ininite long bar o arbitrary cross section an with a circular cylinrical cavity. J. Acoust. Soc. Am. 75(3), , [4] K.Nagaya, Wave propagation in a ro with an arbitrary shape outer bounary an a cylinrical cavity o arbitrary shape. J. Acoust. Soc. Am. 77(5), , [5] K.Nagaya, Direct metho on etermination o eigenrequencies o arbitrary shape plates. ASME.J. Vib. Stress Rel. Des.105, , [6] H. S. Paul, M. Venkatesan, Wave propagation in a pieoelectric ceramic cyliner o arbitrary cross section with a circular cylinrical cavity. J. Acoust. Soc. Am. 85(1), , [7] Vinay Dayal, Longituional waves in a homogeneous anisotrpic cylinrical bars immerse in lui. J. Acoust. Soc. Am. 93(3), , [8] F.Ahma, Giue waves in a transversely isotropic cyliner immerse in a Flui. J. Acoust. Soc. Am.109(3), , 001. [9] B.Nagy, Longituinal guie wave propagation in a transversely isotropic ro Immerse in a lui. J. Acoust. Soc. Am. 98(1), , [10] B.Nagy, H.Nayeh, Viscosity inuce attenuation o longituinal guie Waves in a lui loae ros.j. Acoust. Soc. Am. 100(3), , [11] V.Easwaran, M.L. Munjal, A note on the eect o wall compliance on lowest-orer moe propagation in lui-ille/submerge impeance tubes. J. Acoust. Soc. Am. 9(6), , [1] K.Sinha, J.Plona, S.Kostek an S. Chang, Axisymmetric wave propagation in a lui-loae cylinrical shells. I: Theory; II Theory versus experiment, J. Acoust. Soc. Am. 9, , 199. [13] J.Berliner, R.Solecki, Wave Propagation in a lui-loae, transversely isotropic cyliners. Part I. Analytical ormulation; Part II Numerical results, J. Acoust. Soc. Am. 99, , [14] M.Venkatesan, P.Ponnusamy, Wave propagation in a soli cyliner o arbitrary cross-section immerse in a lui, J. Acoust. Soc. Am. (11), , 00. [15] P.Ponnusamy, R.Selvamani, Dispersion analysis o a generalie magneto thermo elastic cylinrical panel, Journal o thermal stresses. 35(1), , 01. [16] R.Selvamani, P.Ponnusamy, Eect o rotation in an axisymmetric vibration o a transversely isotropic soli bar immerse in an invisci

9 lui, Materials Physics an Mechanics, 15, , 01. [17] G.M.L.Glawell,U.C.Tahbilar, Finite element analysis o the axis symmetric vibrations o cyliners, J.Soun Vibr., ,197. [18] G.R.Buchanan, C.B.Yii, Eect o symmetrical bounary conitions on the vibration o thick hollow cyliners, Appl.Acoust.63, , 00. [19] H.Wang, K.Williams, Vibrational moes o thick cyliner o inite length, J.Soun Vibr.191, , [0] R.K.Singal, K.Wiliiams, A theoretical an experimental stuy o vibrations o thick circular cylinrical shells an rings, J.Vibr, Acoust. Stress Reliab.Des.110, , [1] J.So, A.W.Leissa, Free vibration o thick hollow circular cyliners rom three imensional analysis, J.Vibr, Acoust.119, 89-95, [] D.Zhou et al, 3D vibration analysis o soli an hollow circular cyliners via Chybyshev-Rit metho, Comp. Mthos in Appl.. Mech&Eng.19, ,