In this paper, we have discussed the problem of vibration of cylindrical
shell of aelotropic material of variable density for two different cases-
first, when the density varies linearly and second, when it varies inversely
as the radius vecto

Narain, Surya

Verma, B G

English

IACS Institutional Repository

Radial vibration o f an aeolotropic cylindrical shell of varying density in a magnetic fieldSuBYA Nabain and B. G. Vebma Department of Mathematics, University of Gorakhpur, Gorakhpur.{Received 4 December 1971)In this paper, we have discussed the problem of vibration of cylindrical shell of aolotropio material of variable density for two different cases— first, when the density varies linearly and second, when it varies inversely as the radius vector.1. I ntboditctionYadava (1968) obtained the solution of the problem of vibration of a cylindrical shell in a magnetic field, the material of the shell being aeolotropic and density uniform. In this paper, the discussion has boon extended to the problem of vi- biation of a cylindrical shell of aelotropic material of variable density. Two cases have boon considered. The fiirst, when the density varies linearly and the second, when it varies inversely as the radius vector. Such problems of magneto­elastic vibrations are of much importance in view of increasing investigations on radiation of electromagnetic energy into the vacuum adjacent to magneto-elastic bodies.2. T h e  P b o b le m , F u n d a m en t a l  E q uatio ns and  B o u n d a r y  Co nd it ionsWo consider an aeolotropic, perfectly conducting cylindrical shell of inner and outer radii r^  and r,^  respectively and tlie space outside the shell to be vacuum. We consider the boundary of the shell to be mechanically stress free. Initially there exists an axial magnetic field of intensity H  in the shell. Then the consti­tutive relations for aeolotropic bodies in cylindj'ical coordinates {r, d, z) as given by Love (1944) are,Indian J. Phya, 47p 2 8 7 -2 9 4  (1 9 7 3 )O'rr =  CiiC,-r-l-Ci2600+Ci3ear2( T z t  = -  C 3 iC rr-1-032^98 +  ^ 33^22... (2.1)where o-fr, (Tza and Crr» c^ z are the components of stress and strain res­pectively. The equations of magneto-elasticity for a perfect conductor with unit permeability as deduced by Kailiski (1963) are,<Trr~(TiM + ^  [rotrot(axH )]xf/ =47T(2 .2)287288 Surya Narain and B. G. Verma £=y(^x//). fc = rot(uxH) . . .  (2.3)whore u is the mechanical displaccnient vector, £  the electric intensity vector and h is the perturbation in the magnetic intensity vector.The equations of electromagnetic field in vacuum are,r o t £ »  =  -  Ac dtrot(2.4)(2.5)(2 .6) (2.7)where E*, h* denote the values of quantities E  s,nd h, respectively, in vacuum. For radial vibration, we have,Uq =  Uz — 0, Ur =  Ue"“* (2.8)e_- — ptcat g =  E  e'“ », e „  =  0.r (2.9)Also the other corresponding quantities arc,hr* =- =  0, hz* =  h * =  Fe‘ »«H r  = H q = 0 , B z  =  H ^ (2.10)E r *  =  Eq* =  0, E z *  ^ E * =where U, V, W are functions of r alone. The equation (2.3) gives,I S  =  ^From (2.5) and (2.7) we get,dW  .1  dV__  . ^ 4 . ^ V  =  0dr® r dr r.aPT =  ~  ™CO dr ’(2,12)(2.18)289... (2.14)Radial Vibration etc.The boundary eonditions on the Burface are, ^ o>r+^iT =  Trr* on r =(Trr~\-Trr =  Tj-r* on r —E =  E* on r =E =  E* on r =  rgwhere Trr, Trr* are Maxwell tensors in the shell and vacuum respectively and can be expressed as47t 477 ' or r /Trr*= / ! * = : - &  Fe'®«47T 477Avhile the elastic stress tensor <7fr is expressible as,<r„ =  ) e<“ «3. M ethod  oe S olution  Case 1. Let us assume,(2.15)(2.16)(3.1)where is a constant. The equation (2.2) with the help of (2.1), (2.8), (2.9) and (3.1) becomes,dr^I / l  I __^12 ^21 \ i  __  ____^ 2 _________ l_ _ ^  I____ Pn*^'^ Ur _  Q\ ^Ci i4-Hi’“/4t7 / r dr ^  r^  ^I?utting XI =  we get,'^ 4 -  ) L  =  0(3.2)(3.3)where=  — 4cga+2Cia—2c8ic„+^,=*/477^ ^  4 L 'Ca+ifx^Putting z =  2j3Kr^^  ^ we get,dz^^— T z^ [^   ^ ’5(3.4)whoroand290 Surya Narain and B. G< Verma(3.5)(3.6)A® =Then the solution of (3.4) iswhore ^ , -B arc constants and J„, are Bessel functions of first4-and second kind. Consequently, the solution of (3.2) iswhere A-y, By are constants given byAy =  {\Kfi\A, By =  {lK fi\B ,Making use of recurrence formula,Y^[z) =  y„(s)(Trr, Trr may bo calculated with the help of (3 6). Wo have, where(3.7)4ci.2—(6r?.—3a+2)cii 4and, 2—6n+3a_ t e + 3 a... (3.8)From (2.11) and (3.0) we get,... (S'9)... (3.10)Radial Vibration etc. 291Tho solution of (2.12) with conditions appropriate to the problem is,r = 0 r .  f o r r > r „^  ( t ) f o r r < r i ,where Yq and I q are Bessel functions of order zero. From (2.16) we get,3’rr* =  o n r > r ,From (3.10), (2.13) and (2.10) we have.on r <  rj.E* on r C  r..The boundary conditions (2.14) yieldsJTr,»/»)}+ =  0where/» irlr  ^  ^ I 2~67^+3a \e , +  K [ c ^ = - ^ ) .  0, =  y + — [ -  - 4 — j.(77. ( =  0^xJ»(S D I , { ^ )  =  00 7 ,(  ) =  0.(3.11)(3.13)(3.14)(3.1B)(3.16)Eliminating Ai, C, D from (3.13) to (3.16) wo obtain the frequency equation as,4 ' ^ n i i K r m h ^ - ^ i . - , ,  r „ ( - ^ )292  Surya N arain  and B . G . V erm aCase 2. Let us assume,Q — Po Po -  -(3.17)(3.11)whore is a constant. The equation (2.2) with tho help of (2.1), (2.8), (2.9) and (3.18) becomes,d^Udr^ +/  I I \V  ^ ) 'du_r dra^a ^  I ___ ^ _ oH  ^ r ' H  ^ r« u +  Cii+ 4 ^Putting U = r~^ ij we got as in the previous case,I (^ la a^i) 1 _0dr'^  ~^  . r dr r^  rCiiH- 477Patting z =  2Kr^ we get.where.a*, , W ? ? + / i _ ? l L  =  o * d z^\  z ^ r3 _ C j s ^ ^  v» =  4m»... (3.19)(3.20)(3.21)2 c®“ +  47TRadial Vibration etc. 293The solution of (3.21) is,... (3.22)where A  and B  are constants and </„, Yn are Bessel functions of order n. Conso- quently the solution of (3.19) becomes,V =  ... (3.23)where A ,^ are constants given byAj_ =  {\Kyi^A, B  ^=  {\Kfi\B.Then a-fr and Trr in this case may be calculated with the help of (3.23) as, o-rr =  r3/*‘^-«)[^i{Cn^rU„_,(2Zrl) +  5JJ2Zri)}+B^{c^^KriY^_^{2Kr^ndY^[2Kr^)-\ c-* ... (3.24)where, 8 =  .? c^+CiaandTrr =  J„(2Jrr‘ ) j+^ i|gr*r„_i(2 .g»'t)+  r „ (2 g r l ) j ]e '“ ‘ ... (3.25)E =  “^ H/^-^'><\A^J„(2Kri)+B^YJ,2Kri)\ e‘ “ '. c(3.26)^fr* and JS?* however, remam the gme as in case 1. The boundary concdition (2 14) yields,+  M l  r, D i J  \ =  0An \ c /... (3.27)where.=  («U + ^  )^> </>, =  « +i^{<4i>-«‘ ‘^«-i(2-ff»-**)+?Si,J,(2.ffr,»)}+S,{K»r._i(2^.-,*)+?S,r.(2i-r,»)_  ^  r ,« ‘ -»>'*(7ro| ^  ) =  0 ... (3.28)^ i J „ ( 2 Z r i » ) + B i r , ( 2 & , » ) - ^ ^ f » - « / ‘ D 7 , = 0  . . .  (3.29)4 ,J „(2X r,* )+ S ,r„(2Z r,»)— ■ o r . ( ^ . ) ... (4.30294 Surya Narain & B. G. VermaEliminating A^ , G and D from (3,27), (3.28), (3.29) and (3.30) we get the fre- quency equation as,[ i^>-i*I'«-i(2JCrit)+<5,r,(2ii:r,»)] [?S»V-/«-i(2Jrrii)+^./„(2jrr,»)] ( ^ ) ]+  [ 5  J n W )[ S i  j][r„(2ir,l)]-[^ir,trU2ir>z*)+^,F„(2ii:,r*)]~[,/„(2i:fs«)] A  r^ 'W 'o r .f ... (3.31)RigFlilBlNOESChatterjee I). 1969 Bull Cal Math. Soo. 61, 121 Kailiski S. (1963a) Proc \ibr. Proh. 4Love A. E. H. (1944) A Treatise on Mathematical Theory of Elasticity, Dover Dub. (C.U.P.) Yadava B. (1968) Indian. J. Phys. 42, 643

Radial Vibration of an Aeolotropic Cylindrical Shell of Varying Density in a Magnetic Field

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/2986/1/Radial%20vibration%20of%20an%20aeolotropic%20cylindrical_Surya%20Narain.pdf

Radial Vibration of an Aeolotropic Cylindrical Shell of Varying Density in a Magnetic Field

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository