 World Journal of Mechanics, 2012, 2, 98-112 doi:10.4236/wjm.2012.22012 Published Online April 2012 (http://www.SciRP.org/journal/wjm) 3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method Jagan Nath Sharma, Nivedita Sharma Department of Mathematics, National Institute of Technology, Hamirpur, India Email: jns@nitham.ac.in, niveditanithmr@gmail.com Received December 28, 2011; revised January 29, 2012; accepted February 15, 2012 ABSTRACT This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been un-coupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal mo-tion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal re-laxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use. Keywords: Thermal Relaxation; Matrix Frobenius Method; Toroidal; Poloidal; Hollow Sphere 1. Introduction The theory of thermoelasticity is well established, Nowa- cki . The governing field equations in classical dy- namic coupled thermoelasticity (CT) are wave-type (hy-perbolic) equations of motion and a diffusion-type (para- bolic) equation of heat conduction. Therefore, it is seen that part of the solution of energy equation extends to infinity, implying that if a homogeneous isotropic elastic medium is subjected to thermal or mechanical distur- bances, the effect of temperature and displacement fields is felt at an infinite distance from the source of distur- bance. This shows that part of disturbance has an infinite velocity of propagation, which is physically impossible. With this drawback in mind, Lord and Shulman , Green and Lindsay , modified the Fourier law of heat conduction and constitutive relations so as to get a hy- perbolic equation for heat conduction. These works in- clude the time needed for the acceleration of heat flow and take into account the coupling between temperature and strain fields for isotropic materials. Dhaliwal and Sherief  extended the generalized thermoelasticity  to anisotropic elastic bodies. A wave-like thermal distur- bance is referred as “second sound” by Chandrasekha- raiah . These theories are also supported by experi- ments of Ackerman et al.  that exhibiting the actual occurrence of second sound at low temperatures and small intervals of time. The investigators Singh and Shar- ma  studied the propagation of plane harmonic waves in homogeneous anisotropic heat-conducting elastic ma- terials. Sharma , Sharma and Sharma  presented an exact analysis of the free vibrations of simply supported, homogeneous, transversely isotropic cylindrical panel based on the three-dimensional generalized thermoelas- ticity. The free vibrations of solid and hollow spheres have been the subject of study for a long period, frequently associated with interest in the oscillations of the earth. In the late nineteenth century, Lamb  showed that two basic types of free vibrations namely, 1) the vibrations with zero volume change and zero radial displacement; Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 99and 2) the vibrations with zero radial components of the curl of the displacement, exist in an isotropic sphere. These vibrations are referred as “vibrations of the first and second classes” respectively. Lapwood and Usami  named the first class of vibrations as “torsional or toroidal” and the second class as “spheroidal or poloidal”. Lapwood and Usami  presented an excellent treat- ment of the vibration of a hollow sphere surrounding a liquid core having finite normal and shear rigidity which serves as an approximate model of the Earth. Lamb  derived the equations governing the free vibration of a solid sphere and subsequently Chree  obtained the secular equations of free vibrations of a sphere in the more convenient form. Much later, Sâto and Usami [13, 14] computed and tabulated the natural frequency pa- rameters for an extensive set of modes of vibration for the solid sphere. They provided equations and a com- prehensive set of results for the distribution of displace- ment within the vibrating sphere. Shah et al. [15,16] studied the vibrations of hollow spheres by using two- dimensional theory of elasticity to obtain natural fre- quency parameters. Gupta and Singh  investigated the problems of wave propagation in transradially iso- tropic elastic sphere. They showed that, for a transradi- ally isotropic sphere, the toroidal and spheroidal modes of vibrations are independent of each other. Bargi and Eslami  used Green-Lindsay theory of thermoelasticity to study the thermo-elastic response of functionally graded hollow sphere and investigated the material distribution effects on temperature, displace- ment and stresses. Sharma and Sharma , studied the generalized transradially thermoelastic solid sphere. We have not come across any systematic and exact study on the effect of temperature variations on three di- mensional vibration of heat conducting elastic genera- lized hollow spherical structures. Therefore, the purpose of this paper is to present the exact three dimensional vi- bration analysis of transradially isotropic, thermoelastic generalized hollow sphere subjected to stress free (or rigidly fixed), thermally insulated (or isothermal) boun- dary conditions. The secular equations governing three dimensional vibrations in a generalized hollow sphere have been derived by using Frobenius series method. The derived secular equations for spheroidal (S) modes of vibrations which are dependent on thermal variation, have been solved numerically for zinc and cobalt materials in order to compute lowest frequency and dissipation factor. The obtained results in case of toroidal vibrations are found to be in agreement with those of Cohen et al. . 2. Formulation and Solution We consider the thermoelastic problem for homogeneous, transradially thermally conducting, elastic generalized hollow sphere of inner and outer radii R1 and R2, re- spectively, initially maintained at uniform temperature 0 in the undisturbed state. For generalized spherically isotropic thermoelastic medium, in the spherical polar coordinates T,,,rt, the basic governing equations of motion, heat conduction and constitutive relations can be expressed as follows Sharma and Sharma . ,r,r,r11rsin r12cotrrr rrr r  u   (1) ,,,11rsin r132cotrrrru   (2) ,,,11rsin r13cotrrrru    (3) 3,,1 ,,,222202001 13221cot1sinrr relrrKTT KTTTrrrrCTtTTt eeett     (4) where 11121311 24412111311 24422rr lrrrr lrrcececeTtTcecececeTtTce   (5) 1313333126612, ,1cotsin112sin112cot11 12sinrrrr lrrrrrrrrrcececeTtTuuuce eerrruueurrruuuerrruuuerrruu uerrr     (6) 11112113331313336611122,cc cccccc  2 (7) where rr is stress along radial direction and r, r Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 100 are along tangential direction. Here ,,ruuuu is the displacement vector, is the temperature change, 11 1213 and 44 are five independent isothermal elasticity; 13,,,Tr tc,33,,,cccc and 13,KK are respec- tively, the coefficients of linear thermal expansion and thermal conductivities along and perpendicular to the axis of symmetry,  and eC are the mass density and specific heat at constant strain and and 1 are ther- mal relaxation times respectively 1l0t t is Kronecker’s delta in which for Lord-Shulman (LS) theory and for Green-Lindsay (GL) theory of thermoelasticity. The comma notation is used for spatial derivatives and the superposed dot denotes time differentiation. It can be proved thermodynamically Sharma and Sharma  that 13 and of course 1l0K2l0,K00eC12 1133 1112,cccc0, and 0. We assume in addition that and the isothermal elas- ticity are components of a positive definite fourth-order tensor. The necessary and sufficient conditions for the satisfaction of latter requirement are 0T2212213,cc2rR11 11440,0,ccccrR0, 0,r r (8) 3. Boundary Conditions We consider the free vibrations of a generalized hollow sphere subjected to stress free (or rigidly fixed), ther- mally insulated (or isothermal) boundary conditions at the surfaces 1 (inner radius) and (outer ra- dius). Mathematically, this leads to 1) For stress free, thermally insulated (or isothermal) boundary of the sphere. rrR0, 0,ruu1R,0rTR0,u,0rT2rR (or ) (9a) 0T0Tat (inner radius) and r (outer radius); 1 22) For rigidly fixed, thermally insulated (or isothermal) boundary of the sphere. rr (or ) (9b) at (inner radius) and (outer radius). 4. Solution of the Problem We define the dimensionless quantities 2301144131222 12344 4433 11 12466441133 010144, ,,,,,,,, ,,,, ijii ijesssssTrrRuu RRRcCcccRRRc ccccccccvKTc Rc441144,,,,2TRccKt00,TctT1,cKRTttvtvTvvtcRR      (10) where 244svc and 44 3eCc K are shear wave velocity and characteristic frequency of the generalized hollow sphere respectively. The primes have been sup- pressed for convenience. It is advantageous to express the displacements ,uu and r in terms of functions u,,Gw and de-fined by Sharma and Sharma  as. T1sinGu , 1sinGu, ruw (11) Using Equation (11) in Equations (1)-(4), we find that 22212 122222222 02cc ccrrrrrrt  (12) 222*022 22** 201 2220lKtTrr trr ttwtrrrt  G (13) 2221122222*12312222120lcccGrrrrrtTt Tcccwrr rr    (14) 22132 24222233122*12221111222 0lccccwrrrrrcccc GrrrrTt Trr r2t     (15) where 222221cot sin2  Assume spherical wave solution of the form         10101,,, cos1,,, cos1,,, cos1,,, cositmmnnnitmmnnnitmmnnnitmmnnnrtUrP erwrtWrPerGrtV rPerTrtT rPer     (16) where cosmnP is the Legendre polynomial; n and m are integers and sRv is the dimensionless fre- quency. Upon substituting solution (16) in Equations (12)-(15), we obtain Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 10122221nU 0 (17) 22*420 22**041 02nnnaTnnWV     (18.1) 222232*11110nnnaVcT  212aW (18.2) 2223142 322*1111134 02nnnaacWnncT   V1 (18.3) where 1001102 001,,,llrtiti ti   22 (19) The quantities and 1,2, 3,4iai used in Equa tions (17) and (18) are defined as 12 321212122241232242212322944484144192 24cccacnn c cacnncccaKnnann cc 321 (20) The uncoupling of equations for the displacement po- tential n from n and n indicates the exis- tence of two distinct modes of vibrations. The solution of Equation (11) for corresponds to the Toroidal mode. There is no effect of temperature and generalization on toroidal mode. Toroidal frequencies will be same as ob- tained by Cohen et al. . U,nVW TThe solution of the spherical Bessel Equation (17) is given by   121,nnnnBJ BYUn (21) where 2212192204nncc  and J and Y is Bessel function of first kind and second kind. 1n and 2n are arbitrary constants de-termined from the boundary conditions. B BGeneralized Series Method The system of Equations (18) has been solved with the help of matrix Frobenius method. Clearly the point 0r (i.e. 0) is a regular singular point of Equa- tions (18) and all the coefficients of the differential Equa- tions (18) are finite, single valued and continuous in the interval 12 where 11R and 22R. The quantities satisfy all the necessary conditions to have series expansions and hence the Frobenius power series method is applicable to solve the coupled system of dif- ferential Equations (18). Thus, we have took the solution vector of the type 0sknkkYZ (22) where nnnnYWVT, kkkkZABD, where s is a constant (real or complex) to be determined and ,kkABrR, k are unknown coefficients to be determined. We need solution in the domain 12, 1. The solu- tion (22) is valid in some deleted interval DRrR R00, 2RR (about the origin) where is the radius of convergence. RUpon substituting solution (22) along with its deriva- tives in Equations (18) and simplifying, we get  211200skkkHskH skHZ  (23) where *01, 1,Hdiag, 12,, 1,2,3, ,1,2,3ijilHHskijskHHskilsk   (24) The elements, ijH and ijH, of matrices 1H and 2H are defined as  2211 432123 12213 12222 22233 4*123*13 1**31 0** 232 0,111,,34 ,2412Hskcsk aHsknnc skaHsk cskaHsksk aHsksk aHskHsk skHsk skHnn         (25) Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 102 Equating to zero the coefficients of lowest powers of in Equation (23), we obtain: 2(.. 0)sie100HsZ (26) where 0111ZABD, 1,1,2,ilHsH sil3(27) For the existence of non-trivial solution of Equations (26) we must have 10Hs, which results in the fol- lowing indicial equations 4222400sAsCsa (28) where the coefficients A and C are given by 222342 3411acann cAc, 22 423 141aannaCc The roots of indicial Equations (28) are given as 22221222344,224,AACAACsssa  (29) Clearly the roots j are related through the relation 4152631, 2,3, 4,5, 6sj,,sss ss s, in which 3s is real but the roots 1s and 2s may be, in general, complex. In case the parameter, s, is complex, then leading terms in the complex series solution (22) are of the type: 00 000coslogsin logRIRssssIIABD ZZsis  (30) In order to obtain two independent real solutions, ac-cording to Neuringer , it is sufficient to use any one of the complex root in a part and taking its real and imaginary parts. Also, the treatment of complex case is unlike that of the real root case with the advantage that the differential equation is required to be solved only once in the former case rather than twice as in latter one. For the choice of roots of the indicial equations, the sys-tem of Equations (27) leads to following eigen vectors:   01 041002 051003 0601010001BBZs ZsQsMZs ZsQsMZs ZsM ,, (31) where 22231 4322 22311111, 2,3, 4,5,6jjBjjjcs acsaQs sa nnc saj   and 0A is a constant. Thus we have 000000110 ,120001BBAMBQ QMDM, (32) as the corresponding eigen vectors. Again equating to zero the coefficients of next lowest degree term 1s in Equation (23) and noting that the matrix 1j1Hs is nonsingular for each j, we obtain: 1*11 2011jj0ZHsHsZDZ (33) where 1*11 21,,1,2,3jjDHsHsAil il (34) The matrices 11jHs and 2jHsk can be writ- ten from the Equation (23) by setting and 1ilA are defined in the Appendix A1. Now equating the coefficients of powers of sk equal to zero, we obtain following recurrence relation: 1221210,1,2kkkHskZHsk ZHZk  (35) where the matrices 12,HH and H are defined in Equation (23). This implies that 1212121kjjkkZHskHsk ZHZ  (36) Now putting 0,1,2,3k in Equation (36) suc-cessively and simplifying, we get 21**121*2021kjjkkZ0kHskHsk DHDZDZ   (37) It can be easily shown that the matrix has simi-lar form to that of *2kD12jHsk for even values of and it is alike k22jHs k for odd values of k. Thus we have: **22220 21210,,0, 1,2,3kk kkZDZZDZk  (38) where Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. Copyright © 2012 SciRes. WJM 103**,j3** 1**221221 2331**21122 213322 2121 2kj jkkikj jkkijDHskHskDHDKDHskHskDHDK  (39) Here the elements are given by Equations (A2)-(A3) as defined in the Appendix. Moreover, it can be shown that ,,1,2,ij ijKK ijin Equation (22) is analytic and hence can be differenti-ated term by term. Moreover, the derived series are also analytic functions. Thus the general solution of the system of Equations (18) has the form *222kDok*D*121kDokD, (40) where 61,,,,nnnsjkjk kjkjkjjkoWVTCAsBsDs (42) ***14**101andis33 null matrixDdiagcD (41) where 1,2,3,4,5,6jsj are the eigen-values and ,,kjkj kjAsBsDs are eigenvectors correspond- ing to the eigen-values js and integer . The quanti- ties kjk are arbitrary constants to be evaluated. Conse- quently, the potential functions and T are writ- ten from Equations (12) by using (42) as under: C,wGNoting that both the matrices and *22 0kD*21 0kDlim (kPP, as and using the fact that kk , if each component sequence converges (Cullen ), we can conclude that the series (22) are absolutely and uniformly convergent with in- finite radius of convergence. Therefore, the considered series kP P 2k     162112121,, ,,,,,cos,,,() cossj kim tmnjkk jkjk jnnjkoim tmnnnnwGTr trCAsBsDsPertrBJBYP e     (43) 5.1. Stress-Free Generalized Hollow Sphere The unknowns , 2n and 1nB B1,2,3, 6njkCj4,5, can be evaluated by using boundary conditions (9) at the inner and outer boundaries of the generalized hollow sphere. Upon employing stress free and thermal boundary condi-tions (9a) at the surface 1 and 2 of the sphere and simplifying we have  00010,1,2,3,4,5,6,7,8kil ilnkXXEEil   (44) 5. Secular Dispersion Relation For a generalized spherically isotropic, thermally con- ducting hollow sphere the stress free (or rigidly fixed), thermally insulated (or isothermal) conditions (9) hold. where 0 10203040506012nnnn nnnnXCCCCCCBB (45) 1234561knknknknknknknn2XCC CC CCBB (46)   111103411314 10111011031011051110110711 14(1) 21,for thermally insulated2, for isothermal31,2312sBsssBBccEnncQscsAssk DsPEDs PEsQsAsPPEsQs    P1011sAs PP (47) J. N. SHARMA ET AL. 104   005711115811111133,22nnEJJPPEY Y     PP (48)  111*341131 1411 11111011110311105104(1) 121,for therma lly insulated2,for isothermal112skkBknnskknkskknknccEnncQs cskAsiDsskD sPEDs PE     kP 11111107111 101,5,6,7, 8,and7, 8112skBknkil ilskkBknnskQs APPEEfori lEskQsAPP     (49) Here the elements 011 2,3,4,5,60kllEE lfor k,1 of determinant Equations (47) and (49) can be obtained by just replacing lsl1,3,5,7ilEi2,4,6, 8ilEi1 in with l, while are ob- tained by replacing , 2,3,4,5sl,6 in with 21, 3, 5, 7ilEi. The element il and in (48) can be obtained by replacing Bessel’s function of first kind ,Ei5,6,7,88lJ with that of second kind Y and the elements and l can be obtained by replacing 6, 8ilEi7, 81 in 5,7ilEi and 7,8l with 2 respectively. The Equation (44) holds iff each term vanishes sepa-rately. This implies that 000for =0, 0il XknE (50) 0 for0,0 kkil XknE (51) The Equations (50) and (51) have a non-trivial solu-tion iff 00, ,1,2,3,4,5,6,7,8ilEil for k = 0, (52) 0,,1,2,3,4,5,6,7,8kilEil for (53) 0,0kn After lengthy but straightforward simplifications and reductions, the determinant Equations (52) and (53) lead to the following secular equations. 77 8878870EE EEforn 1 (54)  2d10dnnPmPfor n1where (55) 0det0,,1,2 ,3,4, 5, 60,0ijEil kn (56) det0,,1,2, 3, 4, 5, 60,0kijEilkn (57)   1111034113410 111011031011051110 114(1) 21,for thermally insulated2,for isothermal112ssssBccEnnc csAssDsEDsEskQsAs  (58)    111771111*341131 14 11 11111111311151324(1) 121,for thermally insulated2,for isothermalskkkkskkkskkkBEJJccEnncBs cskAsiDsskD sEDsEQ   k111 11112skksskAs (59) Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 105 The elements of determinant Equations (58) andby just replacing 02,3,4,5,6ilEj (59) can be obtained ,1lslwhile in 1, 3, 5 with ,l,6 are obtained by0ilEi2,42,3,4,5,6sl replacing 10ilEi in 501,ilEi3, with 2. The element Bessel’s function 78E can be ob-of first kind tained by replacingJ with that of second kind Y and th replacing 1e elements 87E can be obtained by in 87E and 88E with 2 respectively. ularThe sec dispersion Equation (54) provides us first class vibrations called Toroidal vibrations (T-modes) as discussed by Cohen et al. . Clearly these modes do ot depend on thermal variations as expected. These are h absence of radial cpheroidal Mosecular Eqesristics of the Sproidal Mode uation (54) provides us nc aracterized by theomponent of dis-placement. 5.1.1. Sde The uations (56) and (57) govern the second class vibrations called Spheroidal vibrations (S-mod) for 0n, 0k and 0,0kn and 0n respec- tively. These relations contain complete informtion re- garding frequency and other charactehe- adal modes of vibrations in a transradially generalized isotropic hollow sphere. The detail of Spheroidal vibra- tions on neglecting the thermal effects and considering only the radial vibrations have been discussed by Ding et al.  in case of elastokinetics. 5.1.2. ToroiThe secular dispersion Eq 23*tan *3*ttt, for 1n (60) 1/2 3/21** *0Jt tJt , for 1n (61) where t*(=h/R) is thickness to mean radius ratio, where thickness of the sphere is defined as 21hR R and mean radius as 212RRR. Clearly these modes do not depend on thermal varia-quations (60) agree with Ding et f radial otropic solid we have tions as expected. The Eal.  and are characterized by the absence ocomponent of displacement. For homogeneous is11 33121344131 32,, ,,.cccccKKK   (62) so that 12n and the secular Equation (61) reduces to  1/21nnnJ tJ3/2** *t t 0 for (63) It can be shown that Equation (63) is identical to the one obtained by Love , [page 284, Equation (38)]. These modes have also been discussed in detail by Cohen et al.  and Ding et al.  and the corresponding frequencies of such Toroidal modes are same as in case of elastokinetics. The analysis in case of coupled ther- moelasticity (CT) can be obtained by setting1n 010tt ) by taking and for uncoupled thermoelasticity (UCT0, 010tt nd all othetheories of dynamd from the ain the present study. Thr relevant results in case of Lic generalized thermoelasbove analysis by takinge secular equ- S and GL ticity can be 1lations aobtaine and 2l, respectively, in Equations (4) and (thly fixed and thermal boundary conditions (9b) at the surface 5) then using e resulting values of these parameters in different rela-tions at various stages. 5.2. Rigidly Fixed Sphere Taking the rigid and 2 rmin1of the sphere on displacements we get following deteantal equations 0, 1,2,3,4,5,6ijdij (64) 0YJ Y 12 21J (65) 2d10dnnPmP (66) here cos, w11111111311111151111,for thermally insulated2,for isothermalskkskkskkskkdAsdBssDsdDs (67) The elements 2,3,4,5,6ildl of determinantal Equation (64) can be obtained by just replacing,1lsl in 1,3,5il l with , 2,3,4,5,6ilsl, while d'2, 4,6il sdi are obtained by replacing 1 in 1,3 ,5ildi with 2. 5.2.1. Toroidal Vibrations Equation (65) corresponds to first class vibrations (To- roidal mode). Clearly these modes do not depend on ther- mal and relaxation time variations as expected. 5.2.2. Spher oidal Vibrations The secular Equations (64) govern the Spheroidal vibra- tions (S-modes) in case of and respectively in a transradGenemoelastic hollow sphere subjected to stress frconditions. These relations contain complete information 0, 0nkial isotropic 0, 1k, nralized ther- ee boundary Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 106 regardin cy ag frequennd other characteristics of the Sphe- roidal modes of vibrations in a transradially isotropic Generalized thermoelastic hollow sphere. 6. Numerical Results and Discussion We consider the case of free vibrations of a transradially, isotropic generalized thermoelastic hollow spheres made up of solid helium and magnesium materials whose physical data is given in Table 1. As given by Shand Dhaliwal and Singeneralplex transcens coplex values of the frequency (ω) giv frequency arma d Sharma  angh . Due to the presence of dissipation term in heat con-duction Equation (4), the secular equations are in comdental equations which provide um- quency (ω). The real part of fre- es us lowest(1/244RRc ) and imaginary part provide us dissipation factor (1/244IDRc), where ReR and ImI, for fixed values of n and k. The computer simulated profiles of lowest frequency and dissi- presented res 1 to 1pation factor D in spheres of solid helium and mag- nesium materials have beenin Figu 2 for different values of thickness to mean radius ratio (t*) in the context of LS, GL, and CT theories of coupled thermoelasticity. The values of the thermal relaxation time has been estimated from Equation (2.5) of Chan- drasekharaiah  02443.izeKvt Cc and 1t has been taken proportional to that of . The sameerical technique 19]. es linearly and re of sphend CT, neralized theorieoelasticity. It has been ob- served that Ω(LS) > Ω(Cee low Heliu0has been used as given by Sharma and Sharma [t numFigures 1 and 3 show the variations of lowest fre- quency (Ω) with thickness to mean radius ratio (t*) for different values of degree of spherical harmonics (n) for solid helium and magnesium respectively in case of stress free generalized hollow sphere. From Figure 3, it is observed that the profile of lowest frequency increases parabolically with the increase of t* and the order with respect to generalized theories of thermoelasticity is Ω(CT) > Ω(GL) >Ω(LS) for n = 1 and n = 2. Figure 3 reveals that the lowest frequency vari- mains dispersionless at all values of the degree- rical harmonics (n) in the context of LS, GL ages of thermthe order for lowest frequency interlaces is T) > Ω(GL) for n = 1 and n = 2. It can be concluded that with the increase of degrof spherical harmonics (n),est frequency increases. From both the Figures we conclude that with the increase of t* lowest frequency increases. Figures 2 and 4 show the variations of damping factor (D) with thickness to mean radius ratio (t*) for different values of degree of spherical harmonics (n) for solid he- Table 1. Physical data for helium and magnesium crystals. QuantityUnitsm Magnesium 11c 2Nm 100.404010 105.974 10 12c 2Nm 100.212010 102.624 10 13c 2Nm 100.0105010 102.17 10 33c 2Nm 100.553010 106.17 10 44c 2Nm 100.1245 10 103.278 10 1 3 21Nm deg21Nm deg62.3620 10 62.641 10 62.68 10 62.68 10 1K 3K 11Wm deg11Wm deg20.3 10 10.2 10 21.7 10 21.7 10  0.04162 0.0202  -1s 131.9890 10 113.58 10 0t s 130.0000091 10 110.27910 00.020.0401 0.0250.055 0.07 85 0.1t* 0.060.080.10.120.140.160.180.20. 0.040.0LSGLCTLSGLCTHollow ball (n = 1) Without ball (n =2)cy (Ω) Lowest equen Figure 1. Variation of lowest frequencwith thickness to mean radius ratio for different values of degree of spherical harmoics (n) for lium material ine of stress free boundary condition. y nhe cas00.010.020.030.040.050.060.070.01 0.0250.04 0.055 0.070.0850.1t*Damping factoer (D)LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2) Figure 2. Variation of damping factor with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for helium material in case of stress free boundary condition. Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 10700.050.10.150.20.250.01 0.025 0.04 0.055 0.070.0850.1t*Lowest equency ( )LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2)Lowest equency (Ω) Figure 3. Variation of lowest frequency with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for magnesium material in case of stress free boundary condition. 00.010.020.030.040.050.060.070.080.010.0250.040.055 0.07 0.0850.1t*Damping factor (D) LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2) Figure 4. Variation of damping factor with thickness to mean radius ratio for different values of degree of sphericam crystals respectively in case of stress free boundary condition. From Figures 2 and 4 it is observed that the trend of profile of damping increases linearly with t* and the order is l harmonics (n) for magnesium material in case of stress free boundary condition. lium and magnesiu DLSDGLDCT  L DCT for n = 2 re- for n = 1 and spectively. An examination is made on the variations of dimen-sionless lowest frequency with respect to thickness to mean radius ratio (t*) ranging from thin spherical shell (t* = 0.05) to the thick spherical shell (t* = 0.1) of iso- tropic materials. The findings confirm that the variation of the Spheroidal frequencies increases with t* and for degree of spherical harmonics (n), the same trend of proickness to mean radius ratio (t*) for ifferent values of degree of spherical harmonics (n) for two materials solid helium and magnesium respectively in case of rigidly fixed boundary condition. For both the materials similar behaviour has been observed i.e. the lowest frequency increases with the increase of t*. In  DLS DG- file for damping has been observed i.e. damping in- creases with the increase of t*. Figures 5 and 7 show the variations of lowest fre- uency () with thqd00.0050.010.0150.020.0250.01 0.025 0.04 0.0550.07 0.0850.1t*Lowest frequency ( )LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2)Lowest frequency (Ω) Figure 5. Variation of lowest frequency with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for helium material in case of rigidly fixed oundary condition. b 00.010.020.01 0.025 0.04 0.0550.07 0.0850.1t*0.030.Damping fa040.050.060.07ctor (D) LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2) Figure 6. Variation of damping factor with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for helium material in case of rigidly fixed boundary condition. 00.0010.0020.0030.0040.0050.0060.0070.0080.009west frequency ( 0.01 0.0250.04 0.055 0.070.0850.1t*Lo )LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2)west frequency (Ω) Lo Figure 7. Variation of lowest frequency with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for magnesium material in case of rigidly fixed boundary condition. Figure 5 the trend of profiles of lowest frequency interlaces is - CTGL CT for n = 1 and n = 2. In Figure 7 the profile of lowest frequency interlaces is CTLS GL in context of linear theories of generalized thermoelasticity. Figures 6 and 8 show the variations of damping factor D) with degree of spherical harmonics (n) for two mate- (Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 108 00.00050.0010.00150.0020.00250.0030.00350.01 0.025 0.040.055 0.07 0.0850.1t*Damping factor (D)LSGLCTLSGLCT Hollow ball (n = 1) Without ball (n =2)of damping factor with thickness to ean radius ratio for different values of degree of spherical terial in case of rigidly fixed boundary condition. rials solid helium and magnesium respectively, for n = 1 and 2. From both the Figures it is observed that the trend of profile of damping factor increases linearly with t*. In Figure 6 the order of damping is D(LS) > D(GL) > D(CT) for n = 2 and the profile of damping interlaces as D(GL) > D(LS) > D(CT) for n = 1. In Figure 8 the order of damping is D(LS) = D(CT) > D(GL) for n = 2 and D(LS) > D(CT) > D(GL) for n = 1. Figures 9 and 11 show the variations of lowest fre-be significantly affected with solid and magnesium respectively. Figure 10 shows tmagnitude of dissipation is constant in the region Figure 8. Variation mharmonics (n) for magnesium maquency  with time ratio for different values of thickness to mean radius ratio (t*) for solid helium and magnesium respectively. The profiles of lowest fre- quency are observed to thermal relaxation time in both the materials with in- creasing values of thickness to mean radius ratio (t*). From both the Figures it is observed that with the in-crease of time ratio lowest frequency is nearly constant value, also lowest frequency has more value for thick shell as compared to thin shells. Figures 10 and 12 show the variations of damping factor (D) with time ratio for different values thickness to mean radius ratio (t*) for heliumhat the 1000tt .2and there is sharp decrease in the region 100.2 0.4tt and then for 100.4tt there is slowing. In Figure 12 there is sharp decrease to decrea se in dampin damping up100.6tt and for 10 0.6 edttreases. From both the figures it is obwith the increase of time ratio the mag-nitude of damping factor is nearly constant value, also lowest frequency has more value for thick shells as com- pared to thin shells. 7. Conclusions zed thermoelastic hollow sphe- has been investigated in the context of LS and GL damping incserv that The effect of thermal variations and thermal relaxation time on lowest frequency and dissipation factor of sphe- rical vibrations in generaliretheories of thermoelasticity with the help of Matrix Frö- benius method. The numerical computations have been done with the help of MATLAB files. It is noticed that the first class vibrations are not af-fected by temperature change, thermal relaxation time and remain independent of rest of the motion. The ob-tained results are similar with the corresponding results 024681012141800.2 0.40.6 0.8116qy()t*= 1/20t*= 1/1010/ttLowest frequency (Ω) Figure 9. Variation of lowest frequency with time ratio for various value of thickness to mean radius ratio (t*) for he-lium material. 0012345Damping FactD)60.2 0.4 0.6 0.81or (t*= 1/20t*= 1/1010/tt Figure 10. Variation of damping factor with time ratio for various value of t* for helium material. 051015202500.2 0.4 0.6 0.81t*= 1/20 t*= 1/10 st frequency (Ω) t1/t0 Lowe Figure 11. Variation of lowest frequency with time ratio for various value of t* for magnesium material. Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. 109g01234500.2 0.4 0.6 0.81Dampinf Factor (D) t*= 1/20t*= 1/1010/ttDamping Factor (D) Figure 12. Variation of damping factor with time ratio for , Lamb  and Cohen et al.  elastokinetics. The lowest frequency of spheroidal vibrations is noticed to be significantly affected due to both temperature variations and relaxation time, in hol- low spheres of both helium and magnesium materials. The lowest frequency and dissipation factor have shown strong dependency on the degree of spherical harmonics (n) and hence the importance of the degree of spherical harmonics must be taken into consideration while de-signing a spherical structure. The effect of relaxation time ratio on lowest frequency and dissipation factor of vibration under consideration is also observed in hollow spheres of helium and m- ies where spherical structures are in frequent use. REFERENCES  W. Nowacki, “Dynamic Problem of ThermoelasticiNoordhoff, Leyden, 1975.  H. W. Lord and Y. Shulman, “The Generalized Dycal Theory of Thermoelasticity,” Journal of Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5various value of t* for magnesium material. obtained by Loveinagnesium materials. This study may find applications in aero- space, navigation, geophysics tribology and other indus- trty,” nami-  A. E. Green and K. A. Lindsay, “ThermoelasticityJournal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689,” .  D. S. Chandrasekharaiah, “Thermoelasticity with SecondSound—A Review,” Applied Mechanics Review, Vol. 39, No. 3, 1986, pp. 355-376. doi:10.1115/1.3143705 R. S. Dhaliwal and R. S. 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SHARMA ET AL. 111Appendix The quantities used in Equations (34) are defined as ,, 1,2,3il jAs il    11122213 12231311 22122121 122113 11232311 22122131313331323333001111 10, 01111 11,10jjjjjj jjjjjjj jjjjjjAAHsHs HsHsAHsHsHs HsAAHs HsHs HsAHsHsHs HsHsAHsHsAHsA11j    (A1) The quantities ,,1,ij jijjKs Ksij2,3 used in Equations (39) are by 22 3111 13*331221 112313 23***22 3212 13*2222 221 122 212221 2122 22221 2122 22122jjjjjjjjjjjjjjjjjHs kHs kKHskHskH skHs kHs kHs kHs kHs kHHs kHs kHskHs kHs kKHskHs kH      1223 32*3321 3121 13*331121 1123 23**22212121 222122 221 122 212221 222 222 21jjjjj jjjjjjjjjjjjjHs kHs kHs ksk HskHskHs kHs kKHskHs kHs kHs kHs kHs kHs kHs kHs kHs k     23*213211233222 13* *22 333113 123322 *3312212222221 221 12221 222122 22122 21jjjjjjjjjjjjjjjjHskHs kHs kHs kHs kHskHs kKHskHs kHs kHs kHs kHs kHs kHsKHskHs kHs k   j 23*21 231123*32 0**21 22121 221 222 21 21jjjjjjjjjjkHs kHs kHskHskHskHskHs kHskHs k     (A2) Copyright © 2012 SciRes. WJM J. N. SHARMA ET AL. Copyright © 2012 SciRes. WJM 112 and   2211 2313 13**2123 112323 **31 3231 3233 332122 22212 221 222221 2122,21 21jjjjjjjjjjjjjjjjHskHsk HskKHskHs kHs kHskHskHsk HskKHs kHskHs kHskKKHs kHsk      (A3)