AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012                 Thermoelastic Damping o...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012                                        ...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012               III. METHOD OF FROBENIUS ...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 where                                  ...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012   Using the boundary conditions (21)-(2...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012where                                   ...
AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012  g   near to the initial guess of ...
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Thermoelastic Damping of Vibrations in a Transversely Isotropic Hollow Cylinder

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The purpose of the paper is to analyze the damping of three-dimensional free vibrations in a transversely isotropic, thermoelastic hollow cylinder, which is initially undeformed
and kept at uniform temperature. The surfaces of the cylinder are subjected to stress free and thermally insulated boundary conditions. The displacement potential functions have been introduced for decoupling the purely shear and longitudinal motions in the equations of motion and heat equation. The purely transverse wave gets decoupled from rest of the motion and is not affected by thermal field. By using the method of separation of variables, the system of governing partial differential equations is reduced to four second order coupled ordinary differential equation in radial coordinate. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. In order to illustrate the analytic results, the numerical solution of various relations and equations are carried out to compute lowest frequency and thermoelastic damping factor with M ATLAB software programming for zinc material. The computer simulated results have been presented graphically.

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Thermoelastic Damping of Vibrations in a Transversely Isotropic Hollow Cylinder

  1. 1. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 Thermoelastic Damping of Vibrations in a Transversely Isotropic Hollow Cylinder H Singh1 and J N Sharma2 1 Department of Mathematics, Lyallpur Khalsa College, Jalandhar(PB)- 144001 India 2 Department of Mathematics, National Institute of Technology, Hamirpur(HP)- 177005 India Email: harjitlkc@rediffmail.com Email: jns@nitham.ac.inAbstract-The purpose of the paper is to analyze the damping of are not affected by thermal field. By using the method ofthree-dimensional free vibrations in a transversely isotropic, separation of variables the model of instant vibration problemthermoelastic hollow cylinder, which is initially undeformed is reduced to four second order coupled ordinary differentialand kept at uniform temperature. The surfaces of the cylinder equations in radial coordinates. One of the standardare subjected to stress free and thermally insulated boundary techniques to solve ordinary differential equations withconditions. The displacement potential functions have beenintroduced for decoupling the purely shear and longitudinal variable coefficients is the Frobenius method available inmotions in the equations of motion and heat equation. The literature, see Tomentschger [12]. The secular equation whichpurely transverse wave gets decoupled from rest of the motion governs the three dimensional vibration of hollow cylinderand is not affected by thermal field. By using the method of has been derived by using Matrix Frobenius method. Theseparation of variables, the system of governing partial numerical solution of secular equation has been carried outdifferential equations is reduced to four second order coupled by MATLAB programming to compute lowest frequency andordinary differential equation in radial coordinate. The matrix thermoelastic damping factor which have been presentedFrobenius method of extended power series is employed toobtain the solution of coupled ordinary differential equations graphically with respect to the parameter t L for first twoalong the radial coordinate. In order to illustrate the analyticresults, the numerical solution of various relations and modes of vibrations n  1,2  .equations are carried out to compute lowest frequency andthermoelastic damping factor with M ATLAB software II. FORMULATION OF PROBLEMprogramming for zinc material. The computer simulatedresults have been presented graphically. We consider a homogeneous transversely isotropic, thermal conducting elastic hollow cylinder of length L andKey words: Damping; Frobenius method; Cylinder radius R at uniform temperature T0 in the undisturbed state I. INTRODUCTION initially. The basic governing equation of motion and heat The vibrations in thermoelastic materials have many conduction for three-dimensional linear coupledapplications in various fields of science and technology, homogeneous and transversely isotropic thermoelasticnamely aerospace, atomic physics, thermal power plants, cylinder in cylindrical co-ordinates r ,  , z  system, in thechemical pipes, pressure vessels, offshore, submarine absence of body force and heat source, are given bystructure, civil engineering structure etc. The hollow cylindersare frequently used as structural components and theirvibrations are obviously important for practical design. Theinvestigations of wave propagation in different cylindericalstructures have been carried out by many researchers [1-6].Ponnusamy [7] studied wave propagation in a generalizedthermoelastic solid cylinder of arbitrary cross section. Suhubiand Erbey [8] investigated longitudinal wave propagation inthermoelastic cylinder. Sharma [9] investigated the vibrationsin a thermoelastic cylindrical panel with voids. The theory of elastic vibrations and waves wellestablished; see Graff [10] and Love [11].The objective of the wherepresent paper is to study the three dimensional vibrationanalysis of simply supported, homogeneous transverselyisotropic, hollow cylinder of length ‘L’ and radius ‘R’. Threedisplacement potential functions are employed for solvingthe equation of motion and heat equation. The purelytransverse wavesget decoupled from the rest of motion and© 2012 AMAE 1DOI: 01.IJPIE.02.01.43
  2. 2. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 On using solutions (8) in equations (5) and (6), we get Here we have used the following non-dimensional quantitiesHere u  u r , u , u z  is the displacement vector;T ( r ,  , z , t ) is the temperature change; c11 , c12 , c13and c 44 are elastic constants;  1 ,  3 and  1 ,  3 arethe coefficients of linear thermal expansion and thermalconductivities along and perpendicular to the axis of symmetryrespectively;  and Ce are the mass density and specificheat at constant strain respectively, eij the strain tensor ; ij the stress tensor. The comma notation is used for spatial-derivatives and superimposed dot denotes time derivatives. whereWe introduce potential functions  , G , W as used bySharma [5] Here v2 is the velocity of purely elastic wave in hollowUsing equation (4) in equation (1) we find that G , W , , T cylinder and  * is the characteristic frequency. The equation (9.1) represents purely transverse wave, which is not affectedsatisfies the equations: by the temperature change. The equation (9.1) is a Bessel’s equation and its possible solutions are where k1 2   k12 . Here E 7 and   E 7 are two arbitrary constants, and J n and Yn are Bessel functions for first and second kind and I n and K n are modified Bessel functionswhere for the first and second kind respectively. Generally k12  0 , so we go on with our derivation by taking the form of  We consider the free vibrations of a right circular hollowcylinder subject to traction free and thermally insulated or for k12  0 , the derivation for k12  0 is obviously similar. .isothermal boundary conditions on the surface r  R which Therefore the solution valid in case of hollow cylinder isis simply supported on the edges z  0 and z  L . taken here asTherefore, we assume solution for threedisplacementfunctions and temperature change as© 2012 AMAE 2DOI: 01.IJPIE.02.01.43
  3. 3. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 III. METHOD OF FROBENIUS where In order to solve the coupled system of differentialequations (9.2)-(9.4) we shall use matrix Frobenius method.The differential equations (9.2), (9.3) and (9.4), can be writtenin matrix form as    Here E1 , E 2 , E 3 , E1 , E 2 , E 3 are arbitrary constants to be evaluated by using boundary condition. where Hence where A standard technique for solving ordinary differentialequations is the method of Frobenius, in which the solutionsare in the form of power series. Clearly r  0 is a regularsingular point of the matrix differential equations (13) andhence, we take the solution of type Here, potential functions Z(r) written from (8) by using (17a) aswhere Here, potential function G, W,  written from (8) by usingHere the coefficients L k s  , M k s , N k s  and the (18a) asparameter s (real or complex) are to be determined. By usingmatrix frobenius method the solution (15) becomeswhere Case-II (when the roots of indicial equation are distinct and differ by integer) Thus the general solution of equation (10) has the form where Here E 1 , E 2 , E 3 , E 4 , E 5 , E 6 are arbitrary constants to be evaluated by using boundary condition.. HenceCase-I (when the roots of indicial equation are distinct anddo not differ by integer)Thus the general solution of equation (10) has the form© 2012 AMAE 3DOI: 01.IJPIE.02.01.43
  4. 4. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 where whereHere, potential functions Z(r) written from (8) by using prime has been suppressed for convenience.(17b) as Case: I-Using the equation (18a) in (24), we obtain the temperature gradient and stresses asHere, potential function G, W,  written from (8) by using(31b) as IV. BOUNDARY CONDITIONS We consider the free vibration of hollow–cylinder whichis subjected to two types of boundary condition at lower andupper surface r  R 1 , R 2 (A) MECHANICAL CONDITIONThe surface are assumed to be traction free, so that where(B)THERMAL CONDITIONThe surface are assumed to be thermally insulated whichleads to V. FREQUENCY EQUATION In this section we derive secular equation for thermoelastichollow cylinder, subjected to traction free, and thermallyinsulated / isothermal boundary conditions at lower andupper surface r  R 1 , R 2  .The displacement, temperaturechange and stresses are obtained as© 2012 AMAE 4DOI: 01.IJPIE.02.01.43
  5. 5. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 Using the boundary conditions (21)-(22) and (25a), weget systems of eight simultaneously equations in     E 1 , E 1 , E 2 , E 2 , E 3 , E 3 , E 7 , E 7 , which will have non-trivial solution if the determinant of their co efficientvanishes. This requirement of non trivial solution leads tosecular equation for hollow cylinder. The secular equationare obtained as  while p ij , i  5 , 6, 7 , 8 can be obtained by just replacing t1 in p by t 2 ij R1 q R q where t1  1 an d t 2  2  1  and R R R R q R2  R1  is the thickness to themean radius ratio to R hollow cylinder Case: II-Using the equation (18b) in (24), we obtained the temperature gradient and stresses as© 2012 AMAE 5DOI: 01.IJPIE.02.01.43
  6. 6. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012where while p ij , i  5 , 6, 7 , 8 can be obtained by just R1 q r e p l a c i n g t1 i n p ij b y t 2 w h e r e t1  1 R R R2 q R 2  R 1  and t 2   1 and q  is the thickness R R R to the mean radius ratio to hollow cylinder VI. NUMERICAL RESULTS AND DISCUSSION In order to illustrate and verify results obtained in previous sections, we present some numerical simulation results. For the purpose of numerical computation, we have considered zinc-crystal like material whose physical data is given below (Dhaliwal and Singh [14]). Using the boundary conditions (21)-(22) and (25b), weget systems of eight simultaneously equations in  E 1 , E 2 , E 3 , E 4 , E 5 , E 6 , E 7 , E 7 , which will have non-trivial solution if the determinant of their Co efficientvanishes. This requirement of non trivial solution leads to Due to presence of dissipation term in heat conductionsecular equation for hollow cylinder. The secular equation equation, the frequency equation in general complexare obtained as transcendental equation provides us complex value of frequency (  ). For fixed value of n and k, the lowest frequency   and dissipation factor (D) are defined as where and The thermoelastic damping factor is given by The numerical computation has been carried out for n  0 , k  0 with the help of MATLAB files .The secular equation (26) has been expressed in the form of   g   and the fixed point iteration numerical technique as outlined in Sharma [15] is used to find approximate solution of© 2012 AMAE 6DOI: 01.IJPIE.02.01.43
  7. 7. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012  g   near to the initial guess of the root with tolerance trends of variations of thermoelastic damping factor for each considered mode of vibrations are almost steady and uniform( 10 5 ). We computed lowest frequency and thermoelastic for of thickness to mean radius ratio of the hollow cylinder. Itdamping factor of first two modes of vibrations n  1 , 2  is also revealed that the variations of thermoelastic damping factor for each mode of vibrations is dispersive in the rangefor different value of parameter axial wave number t L of the and remains close to each other for in case. Moreover, therecylinder. The variations of computer simulated lowest exists atleast one value of parameterforeach at which thefrequency and thermoelastic damping factor are with magnitude of thermoelastic damping factor is same for eachparameter t L in respect to first two modes of vibration in considered mode of vibrations.figures 1 to 2.Fig. 1 represents the variations of lowest CONCLUSIONSfrequency of first two modes of vibrations, versus axial wavenumber for different values of thickness to mean radius ratio The modified Bessel functions and Matrix Frobeniusof the simply supported hollow cylinder of zinc-crystal like method have been successfully used to study the vibrationsmaterial, respectively. It is observed that the lowest frequency of a homogeneous, transversely isotropic hollow cylinderincreases monotonically with axial wave number in both first based on three-dimensional thermoelasticity after decouplingand second modes of vibrations though its magnitude has the equations of motion and heat conduction with the use ofbeen noticed to be larger in second mode of vibrations as potential functions. The decoupled purely transverse modecompared to that of first mode of vibrations. It is also noticed is found to be independent of rest of the motion andthat the magnitude of lowest frequency for each modes of temperature change. The various thermal and mechanicalvibrationsincreases monotonically with increasing value of parameters have significant effects on the natural frequency,q. thermoelastic damping factor of the hollow cylinder. The thermoelastic damping factor increases monotonically with axial wave number but decreases with thickness to mean radius ratio of the hollow cylinder. REFERENCES [1] K. P. Soldatos, V.P. Hadhgeorgian, “Three dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels” J. Sound Vib..” vol. 195, pp-369-384, 1990. [2] A.W. Leissa, I.Y. So, “Free vibrations of thick hollow circular Fig 1. Lowest frequency   of first and second modes cylinders from three dimensional analysis,” ASME J. Vib. Acous., vol. 119, pp. 46-51, 1997. n  1 , 2  versus parameter t L  for different values of [3] I. J. Mirsky, “Wave propagation in transversely isotropic circular thickness to mean radius ratio. cylinder,” J. Acous. Soc. Am.,” vol. 37, pp. 1016-1026, 1965. [4] P.A. Martin, J.R. Berger, “Waves in wood: free vibrations of wooden pole,” J. Mech. Phys. solids, vol. 49, pp. 1155-1178, 2001. [5 ] J.N. Sharma, “Three-dimensional vibration analysis of a homogeneous transversely isotropic cylindrical panel,” J. Acous, Soc, Am., vol. 110, pp. 254-259, 2001. [6 ] J.N. Sharma , P.K. Sharma, “Free vibration ana lysis of homogeneous transversely isotropic thermoelastic cylindrically panel,” J. Therm. stress., vol. 25, pp. 169-182, 2002. [7] P. Ponnusamy, “Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section,” Inter. J. Solids Struct, vol. 44, pp. 5336-5348, 2007. [8 ] E.S. Su hubi, E.S. Erba y, “Longitudina l wave propa gationed Fig 2. Thermoelastic damping factor Q  1  of first and second thermoelastic cylinder,” Journal of Therm. stresses, vol. 9, pp, 279- 295 ,198 6. modes n  1 , 2  versus parameter t L  for different values of [9] P.K.Sharma, D.Kaur, J.N.Sharma, “Three-dimensional vibration thickness to mean radius ratio. analysis of thermoelastic cylindrically panel with voids,” Inter. J. Solids Struct., vol. 45, pp.5049-5048, 2008. From fig. 2 it is observed that the thermoelastic damping [10] K. F. Graff, Wave motion in elastic solids, Dover Publications,factor of each mode of vibrationsincreases monotonically Inc., New York , 2008.with increasing value of axial wave number for different values [11] A. E. H. Love, A Treatise on Mathematical theory of Elasticity, Cambridge: Cambridge University press, 1927.of thickness to mean radius ratio of the hollow cylinder. The© 2012 AMAE 7DOI: 01.IJPIE.02.01.43

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