DERIVING THE TORSIONAL VIBRATION EQUATIONS OF A THREE-LAYER CONICAL SHELL

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1115 DERIVING THE TORSIONAL VIBRATION EQUATIONS OF A THREE-LAYER CONICAL SHELL Sh.N.Isroilov1, Sh.O.Toshbo‘riyev 2 Teachers of the Department of "Natural and Technical Sciences" of Navoi Innovation University (39 Tashkent Street, Karmana District, Navoi Region) Abstract. In this article, the equations of torsional vibration of a three-layer conical shell are derived. The rational construction of a conical shell, from the point of view of its torsional performance, should be such that the main mass of a sufficiently dense material in the form of two layers (bearing layers) is displaced by a certain distance by a third layer (thin wall) consisting of the same or another material, with the third layer being understood as a layer filled with a lighter and less dense material between the two outer layers. Keywords: Three-layer shell, displacements and stresses, deformation tensor. The problem statement. zOr  we consider the torsional vibrations of a three-layer conical shell in a cylindrical coordinate system. We assume that the material of the conical shell is isotropically elastic and homogeneous. Conical shell 0=z inner radius at the tip )1( 0 r through and the outer radius )2( 0 r we mark through lz = inner radius at the tip )1( l r through and the outer radius )2( l r we mark through. In that case, the following relations will be appropriate: .;;; ; ;; 201 )1()4( 01 )1()3()1()2()1( 0 )1( 2010 )4( 0 01 )1( 0 )3( 01 )1( 0 )2( 0 dddrrddrrdrrhtgrr dddrr ddrrdrr lllllll +++=++=+=+= +++= ++=+=  Thicknesses of conical crustal layers 201 ,, ddd we mark through. Since the material of the layers has a viscoelastic property, the integral operator is written as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −−=      −−=   . ; 0 2 0 11 t mmm t mmm dttKtM dttKtL   (1) The projections of the displacement vector of the conical shell points onto the coordinate axes ( ) ( ) ( ) tzrUtzrUtzrU zmmrm ,,,;,,,;,,,   we mark through. Stresses experienced by points of a conical shell )(m ij  and deformations )(m ij  the connections between them are written as follows, involving Bolesmann integrals: ( ) ( ) )()( 1 )( 2m ijm m m m ij ML  += It is known that the differential equations of motion for an arbitrary point of a conical shell in a cylindrical coordinate system are expressed as follows: ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1116            =+   +   +     =+   +   +     = − +   +   +   . 1 ; 21 ; 1 2 2 )()( )( )( 2 2 )( )()()( 2 2)()( )( )( )( t U rzrr t U rzrr t U rzrr zm m zr m zz m z m rz m m r m z mm r rm mm rr m rz m r m rr                     (2) We substitute the expressions of the deformation tensor components and the stress tensor components in the cylindrical coordinate system into the differential equations of motion (2): ; 1 22 1 111 2 2 )(2 )( )( )( 2 )(2 )(2 2 )(2 )()(2 22 )(2 )( 1 t U U U r M r U M r r U rz U M U r U rr U M rr U M r L m r m r m m m r m m r m m m r mm m m r m m m   =                +   −   + +           +   +           +   −   +   +          ; 121 1 2 1 1111 2 )(2 )( )()( 2 )(2 )(2 )( 2 )(2 )( 1 )(2)( 2 )( )( 22 )(2 t U U r M r U M r U M r z U z U r M U U r ML r r U r U r r U r U rr U M m m r m m m m m m m r m m r m m m m m z m r m m m m   =           +−   +           +   + +                   +   +   + +           +   −   −+               (3) . 1 2 11 2 )(2)()( 2 )(2 )( 1 )(2 2 )(2)(2 2 )(2 t U z U r U M r z U M z L z U U r M rzr U r U M m z m z m z m m z m m m m m r m m r m z m   =           +   + +   +   +           +   +           +        The effect of the Laplace operator on a scalar function: . 1111 2 2 2 2 22 2 2 2 22 2 zr r r rr zr rr r  +   +           =   +   +   +   =       The effect of the Laplace operator on a vector function: . 22 2222 →→→→ +         +−+         −−= zz r r r zeUe U rr U Ue U rr U UU              grade z e r e r Udiv z U r U U rr U zr m m z m z m m r m =   +   +   =   ++   +   = →→→ → 1 1)( )()( )( )( )( (4) Taking into account equations (4) it can be written as follows. ( ) ; 2 )( 2 )()( 1t U UMUgraddivML m m m m mm   =++ → →→  (5) To this equation of motion m  longitudinal and mm  , transverse wave potentials ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1117            ++= →→→ z m z mm meroterotgradU  )( we can enter it in the form and get the following. ( ) ; ...... 1                   ++=                   +++                   +++ →→ →→→→ z m z mm z m z mmm z m z mmmm eroterotgrad eroterotgradMeroterotgradgraddivML   or ( ) ( ) ; ...... 11            ++=            +++ +            ++++ →→→→ →→ z m z mm z m z mmmm z m z mmmmmm eroterotgraderoterotMgradM erotegraddivrotMLdgraddivgraML   (6) here .0=            + = →→ z m z m mm erotegraddivrot graddgraddivgra   Taking this into account, equation (6) can be written as follows: ( ) ; ...... 1            ++= =+            +++ →→ →→ z m z mm mm z m z mmmmm eroterotgrad gradMeroterotMgradML   or ( ) ; .... .. 1            ++ +=+            +++ →→ →→ z m z m mmm z m z mmmmm eroterot gradgradMeroterotMgradML   .;; ...... mmmmmmmmm MML  === (7) The displacement vector from the other side z m z m r m r meUeUeUU →→→→ ++= )()()( )(   since we have the following:            −   −   −   =   −   +   =   +   +   = . 11 ; 11 ; 1 2 2 22 2 )( 2 )( 2 )(            mmmm m z mmm m mmm m r rr rrz U rzrr U rzrr U (8) ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1118 If the problem under consideration is axisymmetric, then the displacement vector components of the points of the shell layers are the components of the stress and strain tensors  the angle does not depend on the coordinate. The components of the displacement vector are as follows: . 1 ;; )()( 2 )(           −   =   −=   +   =r r rrz U r U zrr Umm m z m m mm m r   (9) Of time 0t at time a, the conical shell was at rest, and at time c, dynamic loads began to act on its boundary surfaces. Conclusions a) If we consider the torsional vibrations of a three-layer conical elastic shell, the boundary and contact conditions are as follows. Boundary conditions: ( ) ( ) ( ) ( ) .,,, ;,,, )2( 201 )1()2( 201 )1()2( )1()1( 0 )1()1( 0 )1( tzFtzdddrdadddrr tzFtzztgrdaztgrr rlrll rrl     =++++++= =++= (10) Contact terms: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) tzrtzr tzrUtzrUdadrr tzrtzr tzrUtzrUdadrr lrlr llll lrlr llll ,,,, ,,,, ,,,, ,,,, )2()3()3()0( )3( 1 )3( 0 )2()3( )2()1()2()0( )2( 1 )2( 1 )1()2(       = =+= = =+= (11) Initial conditions are zero. When considering axially symmetric torsional vibrations of the shell, the displacements, deformations, and stresses of the layer points do not depend on the angular coordinate. The components of the displacement vector ( ) tzrU i l m,, )()(  non-zero. b) If we consider the longitudinal-radial vibrations of a three-layer conical elastic shell, the boundary and contact conditions will be as follows: Boundary conditions: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) tzFtzrtzFtzrdadddrr tzFtzrtzFtzrdaztgrr rzlrzrlrll rzlrzrlrrl ,,,;,,, ,,,;,,, )2()2()2()2()1()2( 201 )1()2( )1()1()1()1()1()1()1( 0 )1( ==+++= ==+=    (12) Contact terms: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .,,,,;,,,, ;,,,, ,,,,;,,,, ;,,,, )2()3()3()0()3( 2 )3( )3( 1 )3( 0 )2()3( )2()1()2()0()2( 1 )2( )2( 1 )2( 1 )1()2( tzrtzrtzrUtzrU tzrUtzrUdadrr tzrtzrtzrUtzrU tzrUtzrUdadrr lrzlrzlzlz lrlrll lrzlrzlzlz lrlrll       == =+= == =+= (13) Initial conditions are zero. Since axially symmetric longitudinal-radial vibrations of the shell are considered, the layer points ( ) tzrU m r,, )( and ( ) tzrU m z,, )( displacements are nonzero. In this case, the differential equation of motion (2) becomes: ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1119 . ; 2 )(2)()()( 2 )(2 )()( )()( t U rzr t U rzr m z m rr m zz m rz m r mm rr m rz m rr   =+   +     = − +   +         (14) . 1 2 2 2 2 .. .. z rr r M L mmm mmm   +   +   =      = =   (15) d) If we consider the transverse vibrations of a three-layer conical elastic shell, the boundary and contact conditions are as follows. Boundary conditions: ( ) ( ) ( ) ( ) ( ) ( ) .0;0,,, ;,,,,, .0;0,,, ;,,,,, )2()4()2( )2()4()2( 201 )1()4( )2()1()1( )1()1()1()1( 0 )1( == =+++= == =+=       rlrz rlrrll rlrz rlrrl tzr tzFtzrdadddrr tzr tzFtzrdaztgrr (16) Contact terms: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .,,,,,,;,,,,,, ;,,,,,,;,,,,,, ;,,,,,,;,,,,,, .,,,,,,;,,,,,, ;,,,,,,;,,,,,, ;,,,,,,;,,,,,, )3()2()3()0()3()2()3()0( )3()0()3()0()3()2()3()0( )3()2()3()0()3()2()3()0( 0 )2()3( )2()1()2()0()2()1()2()0( )2()1()2()0()2()1()2()0( )2()1()1()0()2()1()2()0( 1 )1()2( tzrtzrtzrUtzrU tzrtzrtzrUtzrU tzrtzrtzrUtzrUdadrr tzrtzrtzrUtzrU tzrtzrtzrUtzrU tzrtzrtzrUtzrUdadrr lzlzlzlz lrlrlrl lrzlrzlrlrll lzlzlzlz lrlrlrl lrzlrzlrlrll           == == ==+= == == ==+= (17) Initial conditions are zero. .0,0,0 =   ==   ==   =ttt m m m m m m       (18) When considering transverse vibrations of the shell, all components of the displacement vector of the layer points are non-zero. 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