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Aeronautics and Aerospace Open Access Journal

Research Article Volume 2 Issue 2

Research on the free vibrational characteristics of isotropic coupled conical-cylindrical shells

Chuang Wu1,2

1State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, China
2Institute of Vibration Shock and Noise, Shanghai Jiao Tong University, China

Correspondence: Chuang Wu, State Key Laboratory of Mechanical System and Vibration, Institute of Vibration Shock and Noise, Shanghai Jiao Tong University, Shanghai, 200240, China

Received: February 05, 2018 | Published: March 7, 2018

Citation: Wu C. Research on the free vibrational characteristics of isotropic coupled conical-cylindrical shells. Aeron Aero Open Access J. 2018;2(2):39-46. DOI: 10.15406/aaoaj.2018.02.00027

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Abstract

Based on the transfer matrix theory and precise integration method, the precise integration transfer matrix method (PITMM) is implemented to investigate the free vibrational characteristics of isotropic coupled conical-cylindrical shells. The influence on the boundary conditions, the shell thickness and the semi-vertex conical angle on the vibrational characteristics are discussed. Based on the Flugge thin shell theory and the transfer matrix method, the field transfer matrix of cylindrical and conical shells is obtained. Taking continuity conditions at the junction of the coupled conical-cylindrical shell into consideration, the field transfer matrix of the coupled shell is constructed. According to the boundary conditions at the ends of the coupled shell, the natural frequencies of the coupled shell are solved by the precise integration method. An approach for studying the free vibrational characteristics of isotropic coupled conical-cylindrical shells is obtained.

Keywords: coupled conical-cylindrical shells, precise integration, transfer matrix, vibration, natural frequency

Introduction

In engineering applications, especially in the field of modern military defence, the cylindrical shells, conical shells and coupled conical-cylindrical shells are basically simplified models of many types of weapons and equipment, such as aircraft, missiles, and submarines. The study of free vibrational characteristics of cylindrical shells is comprehensive. Initially, researchers1-5 investigated cylindrical shells using classic thin shell theories such as Donnell equations, Kennard equations, Flugge equations and Sander-Koiter equations. Harari, Sandman and Laulagnet were representative scholars in the field. Rayleigh6 was a pioneer in the study of free vibrational characteristics of cylindrical shells. The literary work of Leissa7 gave general comments on the free vibrational characteristics of cylindrical shells. The free vibrational characteristics of conical shells with simply-supported boundary conditions are examined using Statistical Energy Analysis by Creenwelge.8 Talebitooti9 and Li10 analysed the free vibrational characteristics of conical shells using the Rayleigh-Ritz method. The kp-Ritz method is used to study conical shells in the work of Liew et al.11 Guo12 applied the multiple factor method to discuss the free vibration characteristics of conical shells.

Unlike the cylindrical shells, the section radius of a conical shell will vary in the axial direction, which increases the complexity and the difficulty in studying conical shells. So far, only an approximate solution for determining the natural frequencies of conical shells has been obtained. Limited work on the analysis of free vibrational characteristics of coupled conical-cylindrical shells has been carried out. Initially, the natural frequencies of the coupled conical-cylindrical shell were solved used FEM. Irie13 investigates the natural frequencies of the coupled shell through the transfer matrix theory. Caresta14 used the two thin theories by Donnell-Mushtari and Flugge to examine the free vibrational characteristics of coupled shells. This paper applies a new method to analyse the free vibrational characteristics of isotropic coupled conical-cylindrical shells, which is different from the approach employed in previous studies. The method is referred to as PITMM. Based on the Flugge thin shell theory, equations of motion for cylindrical and conical shells are derived. The coefficient matrix in the equations of motion for cylindrical and conical shells is calculated using the precise integration method. To take into account the point transfer matrix at the junction of the coupled conical-cylindrical shell and to absorb the matrix assembly solution from FEM, the total transfer matrix of the coupled shell is constructed. According to the boundary conditions, the natural frequencies of the coupled shell are solved.

Equations of motion

Motion of a cylindrical shell
The shell deformation is described by the thin shell theory that is based on linear assumptions. To obtain precise results, the relatively accurate Flugge shell theory is used in this paper. The force balance equation is obtained by analysing the micro-element stress of the cylindrical shell. In this paper, the equations are based on the kinetic theory. Thus, many terms include time items. For the purpose of facilitating the writing and derivation, the dynamic response time item e iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb qcfa4aaWbaaSqabeaajugWaiabgkHiTiaadMgacqaHjpWDcaWG0baa aaaa@3DF9@  is omitted in the remainder of the text. The cylindrical shell coordinates system (γ,φ,χ)s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa Gaeq4SdCMaaiilaiabeA8aQjaacYcacqaHhpWycaGGPaGaam4Caaaa @3F51@  and displacement positive direction are shown in Figure 1.

Based on the Flugge shell theory, the force balance equation of a cylindrical shell is given as follows:

N x x + 1 R N θx θ +ρh ω 2 u=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiabgkGi2kaad6eakmaaBaaaleaajugWaiaadIhaaSqabaaakeaa jugibiabgkGi2kaadIhaaaGaey4kaSIcdaWcaaqaaKqzGeGaaGymaa GcbaqcLbsacaWGsbaaaOWaaSaaaeaajugibiabgkGi2kaad6eakmaa BaaaleaajugWaiabeI7aXjaadIhaaSqabaaakeaajugibiabgkGi2k abeI7aXbaacqGHRaWkcqaHbpGCcaWGObGaeqyYdCNcdaahaaWcbeqa aKqzadGaaGOmaaaajugibiaadwhacqGH9aqpcaaIWaaaaa@574B@         (1)

1 R N ϕ θ + N xθ x Q θ R +ρh ω 2 v=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaamOuaaaakmaalaaabaqcLbsacqGHciIT caWGobGcdaWgaaWcbaqcLbmacqaHvpGzaSqabaaakeaajugibiabgk Gi2kabeI7aXbaacqGHRaWkkmaalaaabaqcLbsacqGHciITcaWGobGc daWgaaWcbaqcLbmacaWG4bGaeqiUdehaleqaaaGcbaqcLbsacqGHci ITcaWG4baaaiabgkHiTOWaaSaaaeaajugibiaadgfakmaaBaaaleaa jugWaiabeI7aXbWcbeaaaOqaaKqzGeGaamOuaaaacqGHRaWkcqaHbp GCcaWGObGaeqyYdCNcdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaa dAhacqGH9aqpcaaIWaaaaa@5F18@     (2)

N θ R + Q x x + 1 R Q θ θ ρh ω 2 w=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaad6eakmaaBaaaleaajugWaiabeI7aXbWcbeaaaOqaaKqzGeGa amOuaaaacqGHRaWkkmaalaaabaqcLbsacqGHciITcaWGrbGcdaWgaa WcbaqcLbmacaWG4baaleqaaaGcbaqcLbsacqGHciITcaWG4baaaiab gUcaROWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOuaaaakmaala aabaqcLbsacqGHciITcaWGrbGcdaWgaaWcbaqcLbmacqaH4oqCaSqa baaakeaajugibiabgkGi2kabeI7aXbaacqGHsislcqaHbpGCcaWGOb GaeqyYdCNcdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaadEhacqGH 9aqpcaaIWaaaaa@5D54@                          (3)             

       Q θ = 1 R M θ θ + M xθ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaaIXaaakeaajugibiaadkfaaaGcdaWcaaqaaKqzGeGaey OaIyRaamytaOWaaSbaaSqaaKqzadGaeqiUdehaleqaaaGcbaqcLbsa cqGHciITcqaH4oqCaaGaey4kaSIcdaWcaaqaaKqzGeGaeyOaIyRaam ytaOWaaSbaaSqaaKqzadGaamiEaiabeI7aXbWcbeaaaOqaaKqzGeGa eyOaIyRaamiEaaaaaaa@5365@        (4)

Q x = M x x + 1 R M θx θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb GcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0JcdaWcaaqa aKqzGeGaeyOaIyRaamytaOWaaSbaaSqaaKqzadGaamiEaaWcbeaaaO qaaKqzGeGaeyOaIyRaamiEaaaacqGHRaWkkmaalaaabaqcLbsacaaI XaaakeaajugibiaadkfaaaGcdaWcaaqaaKqzGeGaeyOaIyRaamytaO WaaSbaaSqaaKqzadGaeqiUdeNaamiEaaWcbeaaaOqaaKqzGeGaeyOa IyRaeqiUdehaaaaa@51F3@     (5)

The Kevin-Kirchhoff membrane forces, shear and all internal forces are

V x = N xθ M xθ R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0JaamOtaOWa aSbaaSqaaKqzadGaamiEaiabeI7aXbWcbeaajugibiabgkHiTOWaaS aaaeaajugibiaad2eakmaaBaaaleaajugWaiaadIhacqaH4oqCaSqa baaakeaajugibiaadkfaaaaaaa@48BF@     (6)

S x = Q x + 1 R M xθ θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0JaamyuaOWa aSbaaSqaaKqzadGaamiEaaWcbeaajugibiabgUcaROWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaamOuaaaakmaalaaabaqcLbsacqGHciIT caWGnbGcdaWgaaWcbaqcLbmacaWG4bGaeqiUdehaleqaaaGcbaqcLb sacqGHciITcqaH4oqCaaaaaa@4D7D@     (7)

N x =D( u x + μ R ( v θ +w)) K R ψ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0Jaamiraiaa cIcakmaalaaabaqcLbsacqGHciITcaWG1baakeaajugibiabgkGi2k aadIhaaaGaey4kaSIcdaWcaaqaaKqzGeGaeqiVd0gakeaajugibiaa dkfaaaGaaiikaOWaaSaaaeaajugibiabgkGi2kaadAhaaOqaaKqzGe GaeyOaIyRaeqiUdehaaiabgUcaRiaadEhacaGGPaGaaiykaiabgkHi TOWaaSaaaeaajugibiaadUeaaOqaaKqzGeGaamOuaaaakmaalaaaba qcLbsacqGHciITcqaHipqEaOqaaKqzGeGaeyOaIyRaamiEaaaaaaa@5CB6@      (8)

N θ =D( 1 R ( v θ +w)+μ u x )+ K R 3 (w+ 2 w θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpcaWGebGa aiikaOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOuaaaacaGGOa GcdaWcaaqaaKqzGeGaeyOaIyRaamODaaGcbaqcLbsacqGHciITcqaH 4oqCaaGaey4kaSIaam4DaiaacMcacqGHRaWkcqaH8oqBkmaalaaaba qcLbsacqGHciITcaWG1baakeaajugibiabgkGi2kaadIhaaaGaaiyk aiabgUcaROWaaSaaaeaajugibiaadUeaaOqaaKqzGeGaamOuaOWaaW baaSqabeaajugWaiaaiodaaaaaaKqzGeGaaiikaiaadEhacqGHRaWk kmaalaaabaqcLbsacqGHciITkmaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaam4DaaGcbaqcLbsacqGHciITcqaH4oqCkmaaCaaaleqabaqc LbmacaaIYaaaaaaajugibiaacMcaaaa@694E@     (9)

N xθ = 1μ 2 D( 1 R u θ + v x )+ K R 2 1μ 2 ( v x ψ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob WcdaWgaaqaaKqzadGaamiEaiabeI7aXbWcbeaajugibiabg2da9OWa aSaaaeaajugibiaaigdacqGHsislcqaH8oqBaOqaaKqzGeGaaGOmaa aacaWGebGaaiikaOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOu aaaakmaalaaabaqcLbsacqGHciITcaWG1baakeaajugibiabgkGi2k abeI7aXbaacqGHRaWkkmaalaaabaqcLbsacqGHciITcaWG2baakeaa jugibiabgkGi2kaadIhaaaGaaiykaiabgUcaROWaaSaaaeaajugibi aadUeaaOqaaKqzGeGaamOuaOWaaWbaaSqabeaajugWaiaaikdaaaaa aOWaaSaaaeaajugibiaaigdacqGHsislcqaH8oqBaOqaaKqzGeGaaG OmaaaacaGGOaGcdaWcaaqaaKqzGeGaeyOaIyRaamODaaGcbaqcLbsa cqGHciITcaWG4baaaiabgkHiTOWaaSaaaeaajugibiabgkGi2kabeI 8a5bGcbaqcLbsacqGHciITcqaH4oqCaaGaaiykaaaa@7003@     (10)

N θx = 1μ 2 D( 1 R u θ + v x )+ K R 2 1μ 2 ( 1 R u θ + ψ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCcaWG4baaleqaaKqzGeGaeyypa0Jc daWcaaqaaKqzGeGaaGymaiabgkHiTiabeY7aTbGcbaqcLbsacaaIYa aaaiaadseacaGGOaGcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWG sbaaaOWaaSaaaeaajugibiabgkGi2kaadwhaaOqaaKqzGeGaeyOaIy RaeqiUdehaaiabgUcaROWaaSaaaeaajugibiabgkGi2kaadAhaaOqa aKqzGeGaeyOaIyRaamiEaaaacaGGPaGaey4kaSIcdaWcaaqaaKqzGe Gaam4saaGcbaqcLbsacaWGsbGcdaahaaWcbeqaaKqzadGaaGOmaaaa aaGcdaWcaaqaaKqzGeGaaGymaiabgkHiTiabeY7aTbGcbaqcLbsaca aIYaaaaiaacIcakmaalaaabaqcLbsacaaIXaaakeaajugibiaadkfa aaGcdaWcaaqaaKqzGeGaeyOaIyRaamyDaaGcbaqcLbsacqGHciITcq aH4oqCaaGaey4kaSIcdaWcaaqaaKqzGeGaeyOaIyRaeqiYdKhakeaa jugibiabgkGi2kabeI7aXbaacaGGPaaaaa@738E@    (11)

M x =K( ψ x + μ R 2 2 w θ 2 1 R u x μ R 2 v θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0Jaam4saiaa cIcakmaalaaabaqcLbsacqGHciITcqaHipqEaOqaaKqzGeGaeyOaIy RaamiEaaaacqGHRaWkkmaalaaabaqcLbsacqaH8oqBaOqaaKqzGeGa amOuaOWaaWbaaSqabeaajugWaiaaikdaaaaaaOWaaSaaaeaajugibi abgkGi2QWaaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWG3baakeaa jugibiabgkGi2kabeI7aXPWaaWbaaSqabeaajugWaiaaikdaaaaaaK qzGeGaeyOeI0IcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWGsbaa aOWaaSaaaeaajugibiabgkGi2kaadwhaaOqaaKqzGeGaeyOaIyRaam iEaaaacqGHsislkmaalaaabaqcLbsacqaH8oqBaOqaaKqzGeGaamOu aOWaaWbaaSqabeaajugWaiaaikdaaaaaaOWaaSaaaeaajugibiabgk Gi2kaadAhaaOqaaKqzGeGaeyOaIyRaeqiUdehaaiaacMcaaaa@6E8E@      (12)

M θ =K( 1 R 2 w+ 1 R 2 2 w θ 2 +μ ψ x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpcaWGlbGa aiikaOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOuaOWaaWbaaS qabeaajugWaiaaikdaaaaaaKqzGeGaam4DaiabgUcaROWaaSaaaeaa jugibiaaigdaaOqaaKqzGeGaamOuaOWaaWbaaSqabeaajugWaiaaik daaaaaaOWaaSaaaeaajugibiabgkGi2QWaaWbaaSqabeaajugWaiaa ikdaaaqcLbsacaWG3baakeaajugibiabgkGi2kabeI7aXPWaaWbaaS qabeaajugWaiaaikdaaaaaaKqzGeGaey4kaSIaeqiVd0McdaWcaaqa aKqzGeGaeyOaIyRaeqiYdKhakeaajugibiabgkGi2kaadIhaaaGaai ykaaaa@6002@       (13)

M xθ = 1μ R K( ψ θ v x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GcdaWgaaWcbaqcLbmacaWG4bGaeqiUdehaleqaaKqzGeGaeyypa0Jc daWcaaqaaKqzGeGaaGymaiabgkHiTiabeY7aTbGcbaqcLbsacaWGsb aaaiaadUeacaGGOaGcdaWcaaqaaKqzGeGaeyOaIyRaeqiYdKhakeaa jugibiabgkGi2kabeI7aXbaacqGHsislkmaalaaabaqcLbsacqGHci ITcaWG2baakeaajugibiabgkGi2kaadIhaaaGaaiykaaaa@5313@         (14)

M θx = 1μ R K( ψ θ + 1 2R u θ 1 2 v x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GcdaWgaaWcbaqcLbmacqaH4oqCcaWG4baaleqaaKqzGeGaeyypa0Jc daWcaaqaaKqzGeGaaGymaiabgkHiTiabeY7aTbGcbaqcLbsacaWGsb aaaiaadUeacaGGOaGcdaWcaaqaaKqzGeGaeyOaIyRaeqiYdKhakeaa jugibiabgkGi2kabeI7aXbaacqGHRaWkkmaalaaabaqcLbsacaaIXa aakeaajugibiaaikdacaWGsbaaaOWaaSaaaeaajugibiabgkGi2kaa dwhaaOqaaKqzGeGaeyOaIyRaeqiUdehaaiabgkHiTOWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaaGOmaaaakmaalaaabaqcLbsacqGHciIT caWG2baakeaajugibiabgkGi2kaadIhaaaGaaiykaaaa@60FC@     (15)

where K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36A6@ and K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36A6@ are the bending rigidity and membrane rigidity, respectively.

K= E h 3 12(1 μ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb GaaeypaOWaaSaaaeaajugibiaadweacaWGObGcdaahaaWcbeqaaKqz adGaaG4maaaaaOqaaKqzGeGaaGymaiaaikdacaGGOaGaaGymaiabgk HiTiabeY7aTPWaaWbaaSqabeaajugWaiaaikdaaaqcLbsacaGGPaaa aaaa@45EE@    (16)

D= Eh 1 μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaeyypa0JcdaWcaaqaaKqzGeGaamyraiaadIgaaOqaaKqzGeGaaGym aiabgkHiTiabeY7aTPWaaWbaaSqabeaajugWaiaaikdaaaaaaaaa@40AC@       (17)

The relationship between the radial displacement and slope is

ψ= w x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qEcqGH9aqpkmaalaaabaqcLbsacqGHciITcaWG3baakeaajugibiab gkGi2kaadIhaaaaaaa@3F40@     (18)

There are sixteen unknown quantities in the above equations. To eliminate eight unknown quantities ( N θ , N xθ , N θx , M θ , M xθ , M θx , Q x , Q θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad6eakmaaBaaaleaajugWaiabeI7aXbWcbeaajugibiaacYca caWGobGcdaWgaaWcbaqcLbmacaWG4bGaeqiUdehaleqaaKqzGeGaai ilaiaad6eakmaaBaaaleaajugWaiabeI7aXjaadIhaaSqabaqcLbsa caGGSaGaamytaOWaaSbaaSqaaKqzadGaeqiUdehaleqaaKqzGeGaai ilaiaad2eakmaaBaaaleaajugWaiaadIhacqaH4oqCaSqabaqcLbsa caGGSaGaamytaOWaaSbaaSqaaKqzadGaeqiUdeNaamiEaaWcbeaaju gibiaacYcacaWGrbGcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGa aiilaiaadgfakmaaBaaaleaajugWaiabeI7aXbWcbeaaaOGaayjkai aawMcaaaaa@63AF@ , eight unknown quantities ( u,v,w,ψ, N x , M x , V x , S x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaadwhacaGGSaGaamODaiaacYcacaWG3bGaaiilaiabeI8a5jaa cYcacaGGobGcdaWgaaWcbaqcLbmacaGG4baaleqaaKqzGeGaaiilai aac2eakmaaBaaaleaajugWaiaacIhaaSqabaqcLbsacaGGSaGaaiOv aOWaaSbaaSqaaKqzadGaaiiEaaWcbeaajugibiaacYcacaGGtbGcda WgaaWcbaqcLbmacaGG4baaleqaaaGccaGLOaGaayzkaaaaaa@5034@ are retained, which are the sectional state vector elements of the cylindrical shell. All quantities are processed into dimensionless quantities and expanded to trigonometric series along the circumferential direction.

(u,w)=h =0 1 n h( u ¯ , w ¯ )sinnθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamyDaiaacYcacaWG3bGaaiykaiabg2da9iaadIgakmaaqahabaWa aabuaeaajugibiaadIgacaGGOaGabmyDayaaraGaaiilaiqadEhaga qeaiaacMcaciGGZbGaaiyAaiaac6gacaWGUbGaeqiUdehaleaajugW aiaad6gaaSqabKqzGeGaeyyeIuoaaSqaaKqzadGaeyOaIyRaeyypa0 JaaGimaaWcbaqcLbmacaaIXaaajugibiabggHiLdaaaa@556A@    (19)

v=h =0 1 n v ¯ cos(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b Gaeyypa0JaamiAaOWaaabCaeaadaaeqbqaaKqzGeGabmODayaaraGa ci4yaiaac+gacaGGZbGaaiikaiaad6gacqaH4oqCcqGHRaWkkmaala aabaqcLbsacqaHXoqycqaHapaCaOqaaKqzGeGaaGOmaaaacaGGPaaa leaajugWaiaad6gaaSqabKqzGeGaeyyeIuoaaSqaaKqzadGaeyOaIy Raeyypa0JaaGimaaWcbaqcLbmacaaIXaaajugibiabggHiLdaaaa@55ED@          (20)

ψ= h R =0 1 n ψ ¯ sin(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qEcqGH9aqpkmaalaaabaqcLbsacaWGObaakeaajugibiaadkfaaaGc daaeWbqaamaaqafabaqcLbsacuaHipqEgaqeaaWcbaqcLbmacaWGUb aaleqajugibiabggHiLdaaleaajugWaiabgkGi2kabg2da9iaaicda aSqaaKqzadGaaGymaaqcLbsacqGHris5aiGacohacaGGPbGaaiOBai aacIcacaWGUbGaeqiUdeNaey4kaSIcdaWcaaqaaKqzGeGaeqySdeMa eqiWdahakeaajugibiaaikdaaaGaaiykaaaa@59B1@       (21)

( N x , N θ , Q x , V x )= K R 2 =0 1 n ( N x ¯ , N θ ¯ , Q x ¯ , V x ¯ )sin(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOtaOWaaSbaaSqaaKqzadGaamiEaaWcbeaajugibiaacYcacaWG obGcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacaGGSaGaamyuaO WaaSbaaSqaaKqzadGaamiEaaWcbeaajugibiaacYcacaWGwbGcdaWg aaWcbaqcLbmacaWG4baaleqaaKqzGeGaaiykaiabg2da9OWaaSaaae aajugibiaadUeaaOqaaKqzGeGaamOuaOWaaWbaaSqabeaajugWaiaa ikdaaaaaaOWaaabCaeaadaaeqbqaaKqzGeGaaiikaOWaa0aaaeaaju gibiaad6eakmaaBaaaleaajugWaiaadIhaaSqabaaaaKqzGeGaaiil aOWaa0aaaeaajugibiaad6eakmaaBaaaleaajugWaiabeI7aXbWcbe aaaaqcLbsacaGGSaGcdaqdaaqaaKqzGeGaamyuaOWaaSbaaSqaaKqz adGaamiEaaWcbeaaaaqcLbsacaGGSaGcdaqdaaqaaKqzGeGaamOvaO WaaSbaaSqaaKqzadGaamiEaaWcbeaaaaqcLbsacaGGPaGaci4Caiaa cMgacaGGUbGaaiikaiaad6gacqaH4oqCcqGHRaWkkmaalaaabaqcLb sacqaHXoqycqaHapaCaOqaaKqzGeGaaGOmaaaacaGGPaaaleaajugW aiaad6gaaSqabKqzGeGaeyyeIuoaaSqaaKqzadGaeyOaIyRaeyypa0 JaaGimaaWcbaqcLbmacaaIXaaajugibiabggHiLdaaaa@8173@        (22)

( N xθ , N θx , Q θ , S x )= K R 2 =0 1 n ( N xθ ¯ , N θx ¯ , Q θ ¯ , S x ¯ )cos(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOtaOWaaSbaaSqaaKqzadGaamiEaiabeI7aXbWcbeaajugibiaa cYcacaWGobGcdaWgaaWcbaqcLbmacqaH4oqCcaWG4baaleqaaKqzGe GaaiilaiaadgfakmaaBaaaleaajugWaiabeI7aXbWcbeaajugibiaa cYcacaWGtbGcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaaiykai abg2da9OWaaSaaaeaajugibiaadUeaaOqaaKqzGeGaamOuaOWaaWba aSqabeaajugWaiaaikdaaaaaaOWaaabCaeaadaaeqbqaaKqzGeGaai ikaOWaa0aaaeaajugibiaad6eakmaaBaaaleaajugWaiaadIhacqaH 4oqCaSqabaaaaKqzGeGaaiilaOWaa0aaaeaajugibiaad6eakmaaBa aaleaajugWaiabeI7aXjaadIhaaSqabaaaaKqzGeGaaiilaOWaa0aa aeaajugibiaadgfakmaaBaaaleaajugWaiabeI7aXbWcbeaaaaqcLb sacaGGSaGcdaqdaaqaaKqzGeGaam4uaOWaaSbaaSqaaKqzadGaamiE aaWcbeaaaaqcLbsacaGGPaGaci4yaiaac+gacaGGZbGaaiikaiaad6 gacqaH4oqCcqGHRaWkkmaalaaabaqcLbsacqaHXoqycqaHapaCaOqa aKqzGeGaaGOmaaaacaGGPaaaleaajugWaiaad6gaaSqabKqzGeGaey yeIuoaaSqaaKqzadGaeyOaIyRaeyypa0JaaGimaaWcbaqcLbmacaaI XaaajugibiabggHiLdaaaa@8840@     (23)

( M x , M θ )= K R =0 1 n ( M x ¯ , M θ ¯ )sin(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamytaOWaaSbaaSqaaKqzadGaamiEaaWcbeaajugibiaacYcacaWG nbGcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacaGGPaGaeyypa0 JcdaWcaaqaaKqzGeGaam4saaGcbaqcLbsacaWGsbaaaOWaaabCaeaa daaeqbqaaKqzGeGaaiikaOWaa0aaaeaajugibiaad2eakmaaBaaale aajugWaiaadIhaaSqabaaaaKqzGeGaaiilaOWaa0aaaeaajugibiaa d2eakmaaBaaaleaajugWaiabeI7aXbWcbeaaaaqcLbsacaGGPaGaci 4CaiaacMgacaGGUbGaaiikaiaad6gacqaH4oqCcqGHRaWkkmaalaaa baqcLbsacqaHXoqycqaHapaCaOqaaKqzGeGaaGOmaaaacaGGPaaale aajugWaiaad6gaaSqabKqzGeGaeyyeIuoaaSqaaKqzadGaeyOaIyRa eyypa0JaaGimaaWcbaqcLbmacaaIXaaajugibiabggHiLdaaaa@6BEC@    (24)

( M xθ , M θx )= K R =0 1 n ( M xθ ¯ , M θx ¯ )cos(nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamytaOWaaSbaaSqaaKqzadGaamiEaiabeI7aXbWcbeaajugibiaa cYcacaWGnbGcdaWgaaWcbaqcLbmacqaH4oqCcaWG4baaleqaaKqzGe Gaaiykaiabg2da9OWaaSaaaeaajugibiaadUeaaOqaaKqzGeGaamOu aaaakmaaqahabaWaaabuaeaajugibiaacIcakmaanaaabaqcLbsaca WGnbGcdaWgaaWcbaqcLbmacaWG4bGaeqiUdehaleqaaaaajugibiaa cYcakmaanaaabaqcLbsacaWGnbGcdaWgaaWcbaqcLbmacqaH4oqCca WG4baaleqaaaaajugibiaacMcaciGGJbGaai4BaiaacohacaGGOaGa amOBaiabeI7aXjabgUcaROWaaSaaaeaajugibiabeg7aHjabec8aWb GcbaqcLbsacaaIYaaaaiaacMcaaSqaaKqzadGaamOBaaWcbeqcLbsa cqGHris5aaWcbaqcLbmacqGHciITcqGH9aqpcaaIWaaaleaajugWai aaigdaaKqzGeGaeyyeIuoaaaa@714D@     (25)

where n is the circumferential modal number. Other dimensionless quantities and dimensionless frequency parameter are

ξ= x l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcqGH9aqpkmaalaaabaqcLbsacaWG4baakeaajugibiaadYgaaaaa aa@3C5E@       (26)

l ¯ = l R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaju gibiaadYgaaaGaeyypa0JcdaWcaaqaaKqzGeGaamiBaaGcbaqcLbsa caWGsbaaaaaa@3B77@      (27)

h ¯ = h R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaju gibiaadIgaaaGaeyypa0JcdaWcaaqaaKqzGeGaamiAaaGcbaqcLbsa caWGsbaaaaaa@3B6F@      (28)

λ 2 = ρh R 2 ω 2 D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBkmaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaeyypa0JcdaWcaaqa aKqzGeGaeqyWdiNaamiAaiaadkfakmaaCaaaleqabaqcLbmacaaIYa aaaKqzGeGaeqyYdCNcdaahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqz GeGaamiraaaaaaa@47FC@     (29)

Through complicated simplification, a first-order matrix differential equation of the cylindrical shell is obtained.

d{ Z(ξ) } dξ = l ¯ U(ξ){ Z(ξ) }+{ F(ξ) }{ p(ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaadsgakmaacmaabaacbmqcLbsacaWFAbGaaiikaiabe67a4jaa cMcaaOGaay5Eaiaaw2haaaqaaKqzGeGaamizaiabe67a4baacqGH9a qpkmaanaaabaqcLbsacaWGSbaaaGqabiaa+vfacaGGOaGaeqOVdGNa aiykaOWaaiWaaeaajugibiaa=PfacaGGOaGaeqOVdGNaaiykaaGcca GL7bGaayzFaaqcLbsacqGHRaWkkmaacmaabaqcLbsacaWFgbGaaiik aiabe67a4jaacMcaaOGaay5Eaiaaw2haaKqzGeGaeyOeI0IcdaGada qaaKqzGeGaa8hCaiaacIcacqaH+oaEcaGGPaaakiaawUhacaGL9baa aaa@5F5C@      (30)

        

where Z(ξ)= { u ¯ v ¯ w ¯ ψ ¯ M x ¯ V ¯ x S ¯ xφ N x ¯ } T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4jaacMcacqGH9aqpkmaacmaabaqcLbsafaqa beqaiaaaaaGcbaqcLbsaceWG1bGbaebaaOqaaKqzGeGabmODayaara aakeaajugibiqadEhagaqeaaGcbaqcLbsacuaHipqEgaqeaaGcbaWa a0aaaeaajugibiaad2eakmaaBaaaleaajugWaiaadIhaaSqabaaaaa GcbaqcLbsaceWGwbGbaebakmaaBaaaleaajugWaiaadIhaaSqabaaa keaajugibiqadofagaqeaOWaaSbaaSqaaKqzadGaamiEaiabeA8aQb WcbeaaaOqaamaanaaabaqcLbsacaWGobGcdaWgaaWcbaqcLbmacaWG 4baaleqaaaaaaaaakiaawUhacaGL9baadaahaaWcbeqaaGqaaKqzad Gaa4hvaaaaaaa@5991@ is the state vector of the cylindrical shell. ( u ¯ , v ¯ , w ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GabmyDayaaraGaaiilaiqadAhagaqeaiaacYcaceWG3bGbaebacaGG Paaaaa@3C57@ are the dimensionless quantities of the axial displacement ( x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3762@ direction), the circumferential displacement ( φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAaaa@3822@ direction) and the radial displacement ( γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@380C@ direction), respectively. ψ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaHip qEgaqeaaaa@384B@ is a dimensionless slope, N x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaju gibiaad6eakmaaBaaaleaajugWaiaadIhaaSqabaaaaaaa@39B5@ is a dimensionless membrane force, M x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaju gibiaad2eakmaaBaaaleaajugWaiaadIhaaSqabaaaaaaa@39B4@ is a dimensionless bending moment, ( V x ¯ , S x ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GcdaqdaaqaaKqzGeGaamOvaOWaaSbaaSqaaKqzadGaamiEaaWcbeaa aaqcLbsaqaaaaaaaaaWdbiaacYcak8aadaqdaaqaaKqzGeGaam4uaO WaaSbaaSqaaKqzadGaamiEaaWcbeaaaaqcLbsacaGGPaaaaa@419A@ are the dimensionless Kelvin-Kirchhoff shear force and shear force, E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb aaaa@372F@ and μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBaaa@381B@ are Young’s modulus and Poisson’s ratio, respectively. Z(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4jaacMcaaaa@3A68@ is the shell element’s state vector and is also a function of the dimensionless variables Z(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4jaacMcaaaa@3A68@ . Z(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4jaacMcaaaa@3A68@ is the coefficient matrix of the differential equation of the cylindrical shell and is an eight-order square matrix. There are 22 non-zero elements in U(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFvbGaaiikaiabe67a4jaacMcaaaa@3A63@ , see Appendix A.

Motion of the conical shell

In a cylindrical coordinate system, the generatrix direction and radial direction of the conical shell are defined as the coordinate direction. The position of any point on a conical shell can be described as (s,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa Gaam4CaabaaaaaaaaapeGaaiila8aacqaH4oqCcaGGPaaaaa@3B4B@ . s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb aaaa@375D@ is length from the top point of the conical shell to any point on the conical shell along the generatrix direction. θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@381B@ is the angle of the point along the circumferential direction in a cylindrical coordinate system. The coordinate system of a conical shell is seen in Figure 2. The analysis of the conical shell force, the force balance equation of a conical shell is given as follows:

1 s (s N s ) s + 1 ssinα N θs θ + N θ s +ρh ω 2 u=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaam4CaaaakmaalaaabaqcLbsacqGHciIT caGGOaGaam4Caiaad6eakmaaBaaaleaajugWaiaadohaaSqabaqcLb sacaGGPaaakeaajugibiabgkGi2kaadohaaaGaey4kaSIcdaWcaaqa aKqzGeGaaGymaaGcbaqcLbsacaWGZbGaci4CaiaacMgacaGGUbGaeq ySdegaaOWaaSaaaeaajugibiabgkGi2kaad6eakmaaBaaaleaajugW aiabeI7aXjaadohaaSqabaaakeaajugibiabgkGi2kabeI7aXbaacq GHRaWkkmaalaaabaqcLbsacaWGobGcdaWgaaWcbaqcLbmacqaH4oqC aSqabaaakeaajugibiaadohaaaGaey4kaSIaeqyWdiNaamiAaiabeM 8a3PWaaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWG1bGaeyypa0Ja aGimaaaa@68BD@     (31)

1 ssinα N θ θ + 1 s N sθ s + N sθ s + Q θ stanα +ρh ω 2 v=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaam4CaiGacohacaGGPbGaaiOBaiabeg7a HbaakmaalaaabaqcLbsacqGHciITcaWGobGcdaWgaaWcbaqcLbmacq aH4oqCaSqabaaakeaajugibiabgkGi2kabeI7aXbaacqGHRaWkkmaa laaabaqcLbsacaaIXaaakeaajugibiaadohaaaGcdaWcaaqaaKqzGe GaeyOaIyRaamOtaOWaaSbaaSqaaKqzadGaam4CaiabeI7aXbWcbeaa aOqaaKqzGeGaeyOaIyRaam4CaaaacqGHRaWkkmaalaaabaqcLbsaca WGobGcdaWgaaWcbaqcLbmacaWGZbGaeqiUdehaleqaaaGcbaqcLbsa caWGZbaaaiabgUcaROWaaSaaaeaajugibiaadgfakmaaBaaaleaaju gWaiabeI7aXbWcbeaaaOqaaKqzGeGaam4CaiGacshacaGGHbGaaiOB aiabeg7aHbaacqGHRaWkcqaHbpGCcaWGObGaeqyYdCNcdaahaaWcbe qaaKqzadGaaGOmaaaajugibiaadAhacqGH9aqpcaaIWaaaaa@731B@     (32)

1 ssinα N θ θ + 1 s N sθ s + N sθ s + Q θ stanα +ρh ω 2 v=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaam4CaiGacohacaGGPbGaaiOBaiabeg7a HbaakmaalaaabaqcLbsacqGHciITcaWGobGcdaWgaaWcbaqcLbmacq aH4oqCaSqabaaakeaajugibiabgkGi2kabeI7aXbaacqGHRaWkkmaa laaabaqcLbsacaaIXaaakeaajugibiaadohaaaGcdaWcaaqaaKqzGe GaeyOaIyRaamOtaOWaaSbaaSqaaKqzadGaam4CaiabeI7aXbWcbeaa aOqaaKqzGeGaeyOaIyRaam4CaaaacqGHRaWkkmaalaaabaqcLbsaca WGobGcdaWgaaWcbaqcLbmacaWGZbGaeqiUdehaleqaaaGcbaqcLbsa caWGZbaaaiabgUcaROWaaSaaaeaajugibiaadgfakmaaBaaaleaaju gWaiabeI7aXbWcbeaaaOqaaKqzGeGaam4CaiGacshacaGGHbGaaiOB aiabeg7aHbaacqGHRaWkcqaHbpGCcaWGObGaeqyYdCNcdaahaaWcbe qaaKqzadGaaGOmaaaajugibiaadAhacqGH9aqpcaaIWaaaaa@731B@        (33)

The Kevin-Kirchhoff membrane forces, shear and all internal forces are

V s = Q s + 1 ssinα M θx θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GcdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaeyypa0JaamyuaOWa aSbaaSqaaKqzadGaam4CaaWcbeaajugibiabgUcaROWaaSaaaeaaju gibiaaigdaaOqaaKqzGeGaam4CaiGacohacaGGPbGaaiOBaiabeg7a HbaakmaalaaabaqcLbsacqGHciITcaWGnbGcdaWgaaWcbaqcLbmacq aH4oqCcaWG4baaleqaaaGcbaqcLbsacqGHciITcqaH4oqCaaaaaa@520E@       (34)

S sθ = N sθ + M θx stanα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GcdaWgaaWcbaqcLbmacaWGZbGaeqiUdehaleqaaKqzGeGaeyypa0Ja amOtaOWaaSbaaSqaaKqzadGaam4CaiabeI7aXbWcbeaajugibiabgU caROWaaSaaaeaajugibiaad2eakmaaBaaaleaajugWaiabeI7aXjaa dIhaaSqabaaakeaajugibiaadohaciGG0bGaaiyyaiaac6gacqaHXo qyaaaaaa@4EEE@       (35)

N s = Eh 1 ν 2 [ u s + v s ( 1 sinα v θ +u+ w tanα ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaeyypa0JcdaWcaaqa aKqzGeGaamyraiaadIgaaOqaaKqzGeGaaGymaiabgkHiTiabe27aUP WaaWbaaSqabeaajugWaiaaikdaaaaaaOWaamWaaeaadaWcaaqaaKqz GeGaeyOaIyRaamyDaaGcbaqcLbsacqGHciITcaWGZbaaaiabgUcaRO WaaSaaaeaajugibiaadAhaaOqaaKqzGeGaam4CaaaakmaabmaabaWa aSaaaeaajugibiaaigdaaOqaaKqzGeGaci4CaiaacMgacaGGUbGaeq ySdegaaOWaaSaaaeaajugibiabgkGi2kaadAhaaOqaaKqzGeGaeyOa IyRaeqiUdehaaiabgUcaRiaadwhacqGHRaWkkmaalaaabaqcLbsaca WG3baakeaajugibiGacshacaGGHbGaaiOBaiabeg7aHbaaaOGaayjk aiaawMcaaaGaay5waiaaw2faaaaa@67E9@     (36)

N θ = Eh 1 ν 2 [ 1 s ( 1 sinα v θ +u+ w tanα )+v u s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGfbGaamiAaaGcbaqcLbsacaaIXaGaeyOeI0IaeqyVd4 McdaahaaWcbeqaaKqzadGaaGOmaaaaaaGcdaWadaqaamaalaaabaqc LbsacaaIXaaakeaajugibiaadohaaaGcdaqadaqaamaalaaabaqcLb sacaaIXaaakeaajugibiGacohacaGGPbGaaiOBaiabeg7aHbaakmaa laaabaqcLbsacqGHciITcaWG2baakeaajugibiabgkGi2kabeI7aXb aacqGHRaWkcaWG1bGaey4kaSIcdaWcaaqaaKqzGeGaam4DaaGcbaqc LbsaciGG0bGaaiyyaiaac6gacqaHXoqyaaaakiaawIcacaGLPaaaju gibiabgUcaRiaadAhakmaalaaabaqcLbsacqGHciITcaWG1baakeaa jugibiabgkGi2kaadohaaaaakiaawUfacaGLDbaaaaa@69FB@     (37)

N θ = Eh 1 ν 2 [ 1 s ( 1 sinα v θ +u+ w tanα )+v u s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGfbGaamiAaaGcbaqcLbsacaaIXaGaeyOeI0IaeqyVd4 McdaahaaWcbeqaaKqzadGaaGOmaaaaaaGcdaWadaqaamaalaaabaqc LbsacaaIXaaakeaajugibiaadohaaaGcdaqadaqaamaalaaabaqcLb sacaaIXaaakeaajugibiGacohacaGGPbGaaiOBaiabeg7aHbaakmaa laaabaqcLbsacqGHciITcaWG2baakeaajugibiabgkGi2kabeI7aXb aacqGHRaWkcaWG1bGaey4kaSIcdaWcaaqaaKqzGeGaam4DaaGcbaqc LbsaciGG0bGaaiyyaiaac6gacqaHXoqyaaaakiaawIcacaGLPaaaju gibiabgUcaRiaadAhakmaalaaabaqcLbsacqGHciITcaWG1baakeaa jugibiabgkGi2kaadohaaaaakiaawUfacaGLDbaaaaa@69FB@     (38)

N θ = Eh 1 ν 2 [ 1 s ( 1 sinα v θ +u+ w tanα )+v u s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGfbGaamiAaaGcbaqcLbsacaaIXaGaeyOeI0IaeqyVd4 McdaahaaWcbeqaaKqzadGaaGOmaaaaaaGcdaWadaqaamaalaaabaqc LbsacaaIXaaakeaajugibiaadohaaaGcdaqadaqaamaalaaabaqcLb sacaaIXaaakeaajugibiGacohacaGGPbGaaiOBaiabeg7aHbaakmaa laaabaqcLbsacqGHciITcaWG2baakeaajugibiabgkGi2kabeI7aXb aacqGHRaWkcaWG1bGaey4kaSIcdaWcaaqaaKqzGeGaam4DaaGcbaqc LbsaciGG0bGaaiyyaiaac6gacqaHXoqyaaaakiaawIcacaGLPaaaju gibiabgUcaRiaadAhakmaalaaabaqcLbsacqGHciITcaWG1baakeaa jugibiabgkGi2kaadohaaaaakiaawUfacaGLDbaaaaa@69FB@    (39)

N θ = Eh 1 ν 2 [ 1 s ( 1 sinα v θ +u+ w tanα )+v u s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGfbGaamiAaaGcbaqcLbsacaaIXaGaeyOeI0IaeqyVd4 McdaahaaWcbeqaaKqzadGaaGOmaaaaaaGcdaWadaqaamaalaaabaqc LbsacaaIXaaakeaajugibiaadohaaaGcdaqadaqaamaalaaabaqcLb sacaaIXaaakeaajugibiGacohacaGGPbGaaiOBaiabeg7aHbaakmaa laaabaqcLbsacqGHciITcaWG2baakeaajugibiabgkGi2kabeI7aXb aacqGHRaWkcaWG1bGaey4kaSIcdaWcaaqaaKqzGeGaam4DaaGcbaqc LbsaciGG0bGaaiyyaiaac6gacqaHXoqyaaaakiaawIcacaGLPaaaju gibiabgUcaRiaadAhakmaalaaabaqcLbsacqGHciITcaWG1baakeaa jugibiabgkGi2kaadohaaaaakiaawUfacaGLDbaaaaa@69FB@    (40)

Q s = 1 s ( M s +s M s s )+ 1 ssinα M θx θ M θ s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb GcdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaeyypa0JcdaWcaaqa aKqzGeGaaGymaaGcbaqcLbsacaWGZbaaaOWaaeWaaeaajugibiaad2 eakmaaBaaaleaajugWaiaadohaaSqabaqcLbsacqGHRaWkcaWGZbGc daWcaaqaaKqzGeGaeyOaIyRaamytaOWaaSbaaSqaaKqzadGaam4Caa WcbeaaaOqaaKqzGeGaeyOaIyRaam4CaaaaaOGaayjkaiaawMcaaKqz GeGaey4kaSIcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWGZbGaci 4CaiaacMgacaGGUbGaeqySdegaaOWaaSaaaeaajugibiabgkGi2kaa d2eakmaaBaaaleaajugWaiabeI7aXjaadIhaaSqabaaakeaajugibi abgkGi2kabeI7aXbaacqGHsislkmaalaaabaqcLbsacaWGnbGcdaWg aaWcbaqcLbmacqaH4oqCaSqabaaakeaajugibiaadohaaaaaaa@68EC@       (41)

Q ϕ = 1 ssinα M θ θ + 1 s ( M sθ +s M sθ s )+ M θs s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb GcdaWgaaWcbaqcLbmacqaHvpGzaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaaIXaaakeaajugibiaadohaciGGZbGaaiyAaiaac6gacq aHXoqyaaGcdaWcaaqaaKqzGeGaeyOaIyRaamytaOWaaSbaaSqaaKqz adGaeqiUdehaleqaaaGcbaqcLbsacqGHciITcqaH4oqCaaGaey4kaS IcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWGZbaaaOWaaeWaaeaa jugibiaad2eakmaaBaaaleaajugWaiaadohacqaH4oqCaSqabaqcLb sacqGHRaWkcaWGZbGcdaWcaaqaaKqzGeGaeyOaIyRaamytaOWaaSba aSqaaKqzadGaam4CaiabeI7aXbWcbeaaaOqaaKqzGeGaeyOaIyRaam 4CaaaaaOGaayjkaiaawMcaaKqzGeGaey4kaSIcdaWcaaqaaKqzGeGa amytaOWaaSbaaSqaaKqzadGaeqiUdeNaam4CaaWcbeaaaOqaaKqzGe Gaam4Caaaaaaa@6D18@     (42)

The relationship between radial displacement and the slope of conical shell satisfies

φ= w s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAcqGH9aqpkmaalaaabaqcLbsacqGHciITcaWG3baakeaajugibiab gkGi2kaadohaaaaaaa@3F2A@      (43)

All quantities are processed into dimensionless quantities and expanded to trigonometric series along the circumferential direction.

( u,w )=h α=0 1 n ( u ˜ , w ˜ )sin( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaadwhacaGGSaGaam4DaaGccaGLOaGaayzkaaqcLbsacqGH9aqp caWGObGcdaaeWbqaamaaqafabaqcLbsacaGGOaGabmyDayaaiaGaai ilaiqadEhagaacaiaacMcaciGGZbGaaiyAaiaac6gakmaabmaabaqc LbsacaWGUbGaeqiUdeNaey4kaSIcdaWcaaqaaKqzGeGaeqySdeMaeq iWdahakeaajugibiaaikdaaaaakiaawIcacaGLPaaaaSqaaKqzadGa amOBaaWcbeqcLbsacqGHris5aaWcbaqcLbmacqaHXoqycqGH9aqpca aIWaaaleaajugWaiaaigdaaKqzGeGaeyyeIuoaaaa@5DD5@         (44)

( u,w )=h α=0 1 n ( u ˜ , w ˜ )sin( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaadwhacaGGSaGaam4DaaGccaGLOaGaayzkaaqcLbsacqGH9aqp caWGObGcdaaeWbqaamaaqafabaqcLbsacaGGOaGabmyDayaaiaGaai ilaiqadEhagaacaiaacMcaciGGZbGaaiyAaiaac6gakmaabmaabaqc LbsacaWGUbGaeqiUdeNaey4kaSIcdaWcaaqaaKqzGeGaeqySdeMaeq iWdahakeaajugibiaaikdaaaaakiaawIcacaGLPaaaaSqaaKqzadGa amOBaaWcbeqcLbsacqGHris5aaWcbaqcLbmacqaHXoqycqGH9aqpca aIWaaaleaajugWaiaaigdaaKqzGeGaeyyeIuoaaaa@5DD5@   (45)

φ= h R α=0 1 n φ ˜ sin( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAcqGH9aqpkmaalaaabaqcLbsacaWGObaakeaajugibiaadkfaaaGc daaeWbqaamaaqafabaqcLbsacuaHgpGAgaacaiGacohacaGGPbGaai OBaOWaaeWaaeaajugibiaad6gacqaH4oqCcqGHRaWkkmaalaaabaqc LbsacqaHXoqycqaHapaCaOqaaKqzGeGaaGOmaaaaaOGaayjkaiaawM caaaWcbaqcLbmacaWGUbaaleqajugibiabggHiLdaaleaajugWaiab eg7aHjabg2da9iaaicdaaSqaaKqzadGaaGymaaqcLbsacqGHris5aa aa@5A92@         (46)

( M s , M θ )= K R α=0 1 n ( M ˜ s , M ˜ θ ) sin( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad2eakmaaBaaaleaajugWaiaadohaaSqabaqcLbsacaGGSaGa amytaOWaaSbaaSqaaKqzadGaeqiUdehaleqaaaGccaGLOaGaayzkaa qcLbsacqGH9aqpkmaalaaabaqcLbsacaWGlbaakeaajugibiaadkfa aaGcdaaeWbqaamaaqafabaWaaeWaaeaajugibiqad2eagaacaOWaaS baaSqaaKqzadGaam4CaaWcbeaajugibiaacYcaceWGnbGbaGaakmaa BaaaleaajugWaiabeI7aXbWcbeaaaOGaayjkaiaawMcaaaWcbaqcLb macaWGUbaaleqajugibiabggHiLdGaci4CaiaacMgacaGGUbGcdaqa daqaaKqzGeGaamOBaiabeI7aXjabgUcaROWaaSaaaeaajugibiabeg 7aHjabec8aWbGcbaqcLbsacaaIYaaaaaGccaGLOaGaayzkaaaaleaa jugWaiabeg7aHjabg2da9iaaicdaaSqaaKqzadGaaGymaaqcLbsacq GHris5aaaa@6B9D@     (47)

( M sθ , M θs )= K R α=0 1 n ( M ˜ sθ , M ˜ θs ) cos( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad2eakmaaBaaaleaajugWaiaadohacqaH4oqCaSqabaqcLbsa caGGSaGaamytaOWaaSbaaSqaaKqzadGaeqiUdeNaam4CaaWcbeaaaO GaayjkaiaawMcaaKqzGeGaeyypa0JcdaWcaaqaaKqzGeGaam4saaGc baqcLbsacaWGsbaaaOWaaabCaeaadaaeqbqaamaabmaabaqcLbsace WGnbGbaGaakmaaBaaaleaajugWaiaadohacqaH4oqCaSqabaqcLbsa caGGSaGabmytayaaiaGcdaWgaaWcbaqcLbmacqaH4oqCcaWGZbaale qaaaGccaGLOaGaayzkaaaaleaajugWaiaad6gaaSqabKqzGeGaeyye IuoaciGGJbGaai4BaiaacohakmaabmaabaqcLbsacaWGUbGaeqiUde Naey4kaSIcdaWcaaqaaKqzGeGaeqySdeMaeqiWdahakeaajugibiaa ikdaaaaakiaawIcacaGLPaaaaSqaaKqzadGaeqySdeMaeyypa0JaaG imaaWcbaqcLbmacaaIXaaajugibiabggHiLdaaaa@70F4@     (48)

( N sθ , N θs , Q θ , S sθ )= K R 2 α=0 1 n ( N ˜ sθ , N ˜ θs , Q ˜ θ , S ˜ sθ ) cos( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad6eakmaaBaaaleaajugWaiaadohacqaH4oqCaSqabaqcLbsa caGGSaGaamOtaOWaaSbaaSqaaKqzadGaeqiUdeNaam4CaaWcbeaaju gibiaacYcacaWGrbGcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsa caGGSaGaam4uaOWaaSbaaSqaaKqzadGaam4CaiabeI7aXbWcbeaaaO GaayjkaiaawMcaaKqzGeGaeyypa0JcdaWcaaqaaKqzGeGaam4saaGc baqcLbsacaWGsbGcdaahaaWcbeqaaKqzadGaaGOmaaaaaaGcdaaeWb qaamaaqafabaWaaeWaaeaajugibiqad6eagaacaOWaaSbaaSqaaKqz adGaam4CaiabeI7aXbWcbeaajugibiaacYcaceWGobGbaGaakmaaBa aaleaajugWaiabeI7aXjaadohaaSqabaqcLbsacaGGSaGabmyuayaa iaGcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsacaGGSaGabm4uay aaiaGcdaWgaaWcbaqcLbmacaWGZbGaeqiUdehaleqaaaGccaGLOaGa ayzkaaaaleaajugWaiaad6gaaSqabKqzGeGaeyyeIuoaciGGJbGaai 4BaiaacohakmaabmaabaqcLbsacaWGUbGaeqiUdeNaey4kaSIcdaWc aaqaaKqzGeGaeqySdeMaeqiWdahakeaajugibiaaikdaaaaakiaawI cacaGLPaaaaSqaaKqzadGaeqySdeMaeyypa0JaaGimaaWcbaqcLbma caaIXaaajugibiabggHiLdaaaa@8A13@     (49)

( N s , N θ , Q s , V s )= K R 2 α=0 1 n ( N ˜ s , N ˜ θ , Q ˜ s , V ˜ s ) sin( nθ+ απ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad6eakmaaBaaaleaajugWaiaadohaaSqabaqcLbsacaGGSaGa amOtaOWaaSbaaSqaaKqzadGaeqiUdehaleqaaKqzGeGaaiilaiaadg fakmaaBaaaleaajugWaiaadohaaSqabaqcLbsacaGGSaGaamOvaOWa aSbaaSqaaKqzadGaam4CaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaey ypa0JcdaWcaaqaaKqzGeGaam4saaGcbaqcLbsacaWGsbGcdaahaaWc beqaaKqzadGaaGOmaaaaaaGcdaaeWbqaamaaqafabaWaaeWaaeaaju gibiqad6eagaacaOWaaSbaaSqaaKqzadGaam4CaaWcbeaajugibiaa cYcaceWGobGbaGaakmaaBaaaleaajugWaiabeI7aXbWcbeaajugibi aacYcaceWGrbGbaGaakmaaBaaaleaajugWaiaadohaaSqabaqcLbsa caGGSaGabmOvayaaiaGcdaWgaaWcbaqcLbmacaWGZbaaleqaaaGcca GLOaGaayzkaaaaleaajugWaiaad6gaaSqabKqzGeGaeyyeIuoaciGG ZbGaaiyAaiaac6gakmaabmaabaqcLbsacaWGUbGaeqiUdeNaey4kaS IcdaWcaaqaaKqzGeGaeqySdeMaeqiWdahakeaajugibiaaikdaaaaa kiaawIcacaGLPaaaaSqaaKqzadGaeqySdeMaeyypa0JaaGimaaWcba qcLbmacaaIXaaajugibiabggHiLdaaaa@7FDA@    (50)

where bending rigidity is K= E h 3 12(1 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb Gaeyypa0JcdaWcaaqaaKqzGeGaamyraiaadIgakmaaCaaaleqabaqc LbmacaaIZaaaaaGcbaqcLbsacaaIXaGaaGOmaiaacIcacaaIXaGaey OeI0IaeqyVd4McdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaacMca aaaaaa@4636@ , Young’s modulus and Poisson’s ratio are E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb aaaa@372F@ and ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH9o GBaaa@381D@ , respectively. n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ is the circumferential modal number. α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqqaaaa aaaaWpa8qacqaHXoqycaqG9aGaaGymaaaa@39E2@ and α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqqaaaa aaaaWpa8qacqaHXoqycaqG9aGaaeimaaaa@39DA@ are the symmetric or the anti-symmetric modal, respectively. R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb aaaa@373C@ is the radius at the larger end of the conical shell. h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb aaaa@3752@ is the thickness of the conical shell. Other dimensionless quantities are presented as

ξ= s R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcqGH9aqpkmaalaaabaqcLbsacaWGZbaakeaajugibiaadkfaaaaa aa@3C3F@     (51)

ξ 1 = L start R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEkmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGmbGcdaWgaaWcbaqcLbmacaWGZbGaamiDaiaadggaca WGYbGaamiDaaWcbeaaaOqaaKqzGeGaamOuaaaaaaa@4507@    (52)

ξ 2 = L end R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEkmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpkmaalaaa baqcLbsacaWGmbGcdaWgaaWcbaqcLbmacaWGLbGaamOBaiaadsgaaS qabaaakeaajugibiaadkfaaaaaaa@4307@      (53)

h ˜ = h R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGOb GbaGaacqGH9aqpkmaalaaabaqcLbsacaWGObaakeaajugibiaadkfa aaaaaa@3B6D@        (54)

λ 2 = ρh R 2 ω 2 D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBkmaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaeyypa0JcdaWcaaqa aKqzGeGaeqyWdiNaamiAaiaadkfakmaaCaaaleqabaqcLbmacaaIYa aaaKqzGeGaeqyYdCNcdaahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqz GeGaamiraaaaaaa@47FC@     (55)

s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb aaaa@375D@ , L start MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb GcdaWgaaWcbaqcLbmacaWGZbGaamiDaiaadggacaWGYbGaamiDaaWc beaaaaa@3D6C@ , L end MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb GcdaWgaaWcbaqcLbmacaWGLbGaamOBaiaadsgaaSqabaaaaa@3B6B@ are described in \. ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCaaa@3825@ , ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHjp WDaaa@3832@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3819@ are, respectively, material density, circular frequency and the dimensionless frequency parameter. There are sixteen unknown quantities in the above equations. To eliminate eight unknown quantities ( M θ , M sθ , M θs , N sθ , N θs , Q θ , N θ , Q s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamytaOWaaSbaaSqaaKqzadGaeqiUdehaleqaaKqzGeaeaaaaaaaa a8qacaGGSaWdaiaad2eakmaaBaaaleaajugWaiaadohacqaH4oqCaS qabaqcLbsapeGaaiila8aacaWGnbGcdaWgaaWcbaqcLbmacqaH4oqC caWGZbaaleqaaKqzGeWdbiaacYcapaGaamOtaOWaaSbaaSqaaKqzad Gaam4CaiabeI7aXbWcbeaajugib8qacaGGSaWdaiaad6eakmaaBaaa leaajugWaiabeI7aXjaadohaaSqabaqcLbsapeGaaiila8aacaWGrb GcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsapeGaaiila8aacaWG obGcdaWgaaWcbaqcLbmacqaH4oqCaSqabaqcLbsapeGaaiila8aaca WGrbGcdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaaiykaaaa@64D4@ , eight unknown quantities (u,v,w,φ, M s , V s , N s , S sθ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamyDaabaaaaaaaaapeGaaiila8aacaWG2bWdbiaacYcapaGaam4D a8qacaGGSaWdaiabeA8aQ9qacaGGSaWdaiaad2eakmaaBaaaleaaju gWaiaadohaaSqabaqcLbsapeGaaiila8aacaWGwbGcdaWgaaWcbaqc LbmacaWGZbaaleqaaKqzGeWdbiaacYcapaGaamOtaOWaaSbaaSqaaK qzadGaam4CaaWcbeaajugib8qacaGGSaWdaiaadofakmaaBaaaleaa jugWaiaadohacqaH4oqCaSqabaqcLbsacaGGPaaaaa@530B@ are retained, which are the sectional state vector elements of the conical shell. Then, the first-order matrix differential equation of the conical shell is obtained.

d{ Z(ξ) } dξ =U(ξ){ Z(ξ) }+{ F(ξ) }{ p(ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaadsgakmaacmaabaacbmqcLbsacaWFAbGaaiikaiabe67a4jaa cMcaaOGaay5Eaiaaw2haaaqaaKqzGeGaamizaiabe67a4baacqGH9a qpieqacaGFvbGaaiikaiabe67a4jaacMcakmaacmaabaqcLbsacaWF AbGaaiikaiabe67a4jaacMcaaOGaay5Eaiaaw2haaKqzGeGaey4kaS IcdaGadaqaaKqzGeGaa8NraiaacIcacqaH+oaEcaGGPaaakiaawUha caGL9baajugibiabgkHiTOWaaiWaaeaajugibiaa=bhacaGGOaGaeq OVdGNaaiykaaGccaGL7bGaayzFaaaaaa@5DC1@

where { Z(ξ) }= { u ˜ , v ˜ , w ˜ , φ ˜ , M ˜ s , V ˜ s , N ˜ s , S ˜ sθ } T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaie Wajugibiaa=PfacaGGOaGaeqOVdGNaaiykaaGccaGL7bGaayzFaaqc LbsacqGH9aqpkmaacmaabaqcLbsaceWG1bGbaGaacaGGSaGabmODay aaiaGaaiilaiqadEhagaacaiaacYcacuaHgpGAgaacaiaacYcaceWG nbGbaGaakmaaBaaaleaajugWaiaadohaaSqabaqcLbsacaGGSaGabm OvayaaiaGcdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaaiilaiqa d6eagaacaOWaaSbaaSqaaKqzadGaam4CaaWcbeaajugibiaacYcace WGtbGbaGaakmaaBaaaleaajugWaiaadohacqaH4oqCaSqabaaakiaa wUhacaGL9baadaahaaWcbeqaaKqzadGaamivaaaaaaa@5D8D@ is the state vector of the conical shell. U(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbsaca WFvbGaaiikaiabe67a4jaacMcaaaa@3A61@ is the variable coefficient matrix, { F(ξ) }{ p(ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaie Wajugibiaa=zeacaGGOaGaeqOVdGNaaiykaaGccaGL7bGaayzFaaqc LbsacqGHsislkmaacmaabaqcLbsacaWFWbGaaiikaiabe67a4jaacM caaOGaay5Eaiaaw2haaaaa@44EC@ are exciting loads. The non-zero elements in U(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbsaca WFvbGaaiikaiabe67a4jaacMcaaaa@3A61@ are shown in Appendix B.

Figure 1 Coordinate system of a cylindrical shell.
Figure 2 Coordinate system of the conical shell

Solutions to equations

Assuming that the exciting loads of Eqs. (30), (56) are zero, the equations of motion are simplified to

d{ Z cy (ξ) } dξ = U cy (ξ){ Z cy (ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaadsgakmaacmaabaacbmqcLbsacaWFAbGcdaWgaaWcbaqcLbma caWGJbGaamyEaaWcbeaajugibiaacIcacqaH+oaEcaGGPaaakiaawU hacaGL9baaaeaajugibiaadsgacqaH+oaEaaGaeyypa0dcbeGaa4xv aOWaaSbaaSqaaKqzadGaam4yaiaadMhaaSqabaqcLbsacaGGOaGaeq OVdGNaaiykaOWaaiWaaeaajugibiaa=PfakmaaBaaaleaajugWaiaa dogacaWG5baaleqaaKqzGeGaaiikaiabe67a4jaacMcaaOGaay5Eai aaw2haaaaa@58E8@      (57)

d{ Z co (ξ) } dξ = U co (ξ){ Z co (ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaadsgakmaacmaabaacbmqcLbsacaWFAbGcdaWgaaWcbaqcLbma caWGJbGaam4BaaWcbeaajugibiaacIcacqaH+oaEcaGGPaaakiaawU hacaGL9baaaeaajugibiaadsgacqaH+oaEaaGaeyypa0dcbeGaa4xv aOWaaSbaaSqaaKqzadGaam4yaiaad+gaaSqabaqcLbsacaGGOaGaeq OVdGNaaiykaOWaaiWaaeaajugibiaa=PfakmaaBaaaleaajugWaiaa dogacaWGVbaaleqaaKqzGeGaaiikaiabe67a4jaacMcaaOGaay5Eai aaw2haaaaa@58CA@     (58)

Eqs. (57) and (58) are the equations of motion for the cylindrical and conical shell, respectively, which are dealt with as follows

d{ Z(ξ) } dξ =U(ξ){ Z(ξ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaju gibiaadsgakmaacmaabaacbmqcLbsacaWFAbGaaiikaiabe67a4jaa cMcaaOGaay5Eaiaaw2haaaqaaKqzGeGaamizaiabe67a4baacqGH9a qpieqacaGFvbGaaiikaiabe67a4jaacMcakmaacmaabaqcLbsacaWF AbGaaiikaiabe67a4jaacMcaaOGaay5Eaiaaw2haaaaa@4D3C@       (59)

ξ 1 ξ d{ Z(ξ) } { Z(ξ) } = ξ 1 ξ U(τ) dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada WcaaqaaKqzGeGaamizaOWaaiWaaeaaieWajugibiaa=PfacaGGOaGa eqOVdGNaaiykaaGccaGL7bGaayzFaaaabaWaaiWaaeaajugibiaa=P facaGGOaGaeqOVdGNaaiykaaGccaGL7bGaayzFaaaaaaWcbaqcLbma cqaH+oaElmaaBaaameaajugWaiaaigdaaWqabaaaleaajugWaiabe6 7a4bqcLbsacqGHRiI8aiabg2da9OWaa8qCaeaaieqajugibiaa+vfa caGGOaGaeqiXdqNaaiykaaWcbaqcLbmacqaH+oaElmaaBaaameaaju gWaiaaigdaaWqabaaaleaajugWaiabe67a4bqcLbsacqGHRiI8aiaa dsgacqaHepaDaaa@634A@        (60)

lnZ(ξ)| ξ 1 ξ = ξ 1 ξ U(τ) dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqGaaeaaju gibiGacYgacaGGUbacbmGaa8NwaiaacIcacqaH+oaEcaGGPaaakiaa wIa7amaaDaaaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaaGymaa adbeaaaSqaaKqzadGaeqOVdGhaaKqzGeGaeyypa0JcdaWdXbqaaGqa bKqzGeGaa4xvaiaacIcacqaHepaDcaGGPaaaleaajugWaiabe67a4T WaaSbaaWqaaKqzadGaaGymaaadbeaaaSqaaKqzadGaeqOVdGhajugi biabgUIiYdGaamizaiabes8a0baa@5A11@      (61)

ln( Z(ξ) Z( ξ 1 ) )= ξ 1 ξ U(τ) dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaOWaaeWaaeaadaWcaaqaaGqadKqzGeGaa8NwaiaacIcacqaH +oaEcaGGPaaakeaajugibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWcba qcLbsacaaIXaaaleqaaKqzGeGaaiykaaaaaOGaayjkaiaawMcaaKqz GeGaeyypa0JcdaWdXbqaaGqabKqzGeGaa4xvaiaacIcacqaHepaDca GGPaaaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaaGymaaadbeaa aSqaaKqzadGaeqOVdGhajugibiabgUIiYdGaamizaiabes8a0baa@5910@         (62)

Z(ξ)=exp( ξ 1 ξ U(τ) dτ)Z( ξ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4jaacMcacqGH9aqpciGGLbGaaiiEaiaaccha caGGOaGcdaWdXbqaaGqabKqzGeGaa4xvaiaacIcacqaHepaDcaGGPa aaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaaGymaaadbeaaaSqa aKqzadGaeqOVdGhajugibiabgUIiYdGaamizaiabes8a0jaacMcaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzGeGaaGymaaWcbeaajugi biaacMcaaaa@57FC@      (63)

In the following chapters, the solution for the coefficient matrix exp( ξ 1 ξ U(τ) dτ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGLb GaaiiEaiaacchacaGGOaGcdaWdXbqaaGqabKqzGeGaa8xvaiaacIca cqaHepaDcaGGPaaaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaaG ymaaadbeaaaSqaaKqzadGaeqOVdGhajugibiabgUIiYdGaamizaiab es8a0jaacMcaaaa@4CE3@ using the precise integration method is presented.

Solutions for the coefficient matrix of the cylindrical shell

In the numerical calculation, the cylindrical shell is divided into a series of segments. The node coordinate of a segment is exp( ξ 1 ξ U(τ) dτ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGLb GaaiiEaiaacchacaGGOaGcdaWdXbqaaGqabKqzGeGaa8xvaiaacIca cqaHepaDcaGGPaaaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaaG ymaaadbeaaaSqaaKqzadGaeqOVdGhajugibiabgUIiYdGaamizaiab es8a0jaacMcaaaa@4CE3@ , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36C6@ =i+1,i+2,i+3,…. Any coordinates of contiguous nodes are k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3755@ and ξ k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEkmaaBaaaleaajugWaiaadUgacqGHRaWkcaaIXaaaleqaaaaa@3C24@ , where ξ k+1 = ξ k +Δξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEkmaaBaaaleaajugWaiaadUgacqGHRaWkcaaIXaaaleqaaKqzGeGa eyypa0JaeqOVdGNcdaWgaaWcbaqcLbmacaWGRbaaleqaaKqzGeGaey 4kaSIaeyiLdqKaeqOVdGhaaa@4676@ . The coefficient matrix U(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbsaca WFvbGaaiikaiabe67a4jaacMcaaaa@3A61@ for the cylindrical shell is independent of ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEaaa@3828@ . Thus, the coefficient matrix in Eq. (63) can be written as

e UΔξ =exp( ξ k ξ k+1 U(τ) dτ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb GcdaahaaWcbeqaaGqadKqzadGaa8xvaiabfs5aejabe67a4baajugi biabg2da9iGacwgacaGG4bGaaiiCaiaacIcakmaapehabaacbeqcLb sacaGFvbGaaiikaiabes8a0jaacMcaaSqaaKqzadGaeqOVdG3cdaWg aaadbaqcLbmacaWGRbaameqaaaWcbaqcLbmacqaH+oaElmaaBaaame aajugWaiaadUgacqGHRaWkcaaIXaaameqaaaqcLbsacqGHRiI8aiaa dsgacqaHepaDcaGGPaaaaa@5905@      (64)

Assuming that

Φ 0 (Δξ)= e UΔξ =exp ( H ) 2 s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccmqcLbsacq WFMoGrkmaaBaaaleaajugWaiaaicdaaSqabaqcLbsacaGGOaGaeuiL dqKaeqOVdGNaaiykaiabg2da9iaadwgakmaaCaaaleqabaacbmqcLb macaGFvbGaeuiLdqKaeqOVdGhaaKqzGeGaeyypa0JaciyzaiaacIha caGGWbGcdaqadaqaaKqzGeGaa4hsaaGccaGLOaGaayzkaaWaaWbaaS qabeaajugWaiaaikdakmaaCaaameqabaqcLbmacaWGZbaaaaaaaaa@5258@              (65)

where H=U Δξ 2 s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFibGaeyypa0Jaa8xvaOWaaSaaaeaajugibiabfs5aejabe67a4bGc baqcLbsacaaIYaGcdaahaaWcbeqaaKqzadGaam4Caaaaaaaaaa@409A@ , and a value of 20 is recommended for s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb aaaa@375D@ . s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb aaaa@375D@ can be expressed in terms of the Taylor series by

exp( H )= I 8 + k=1 H k k! = I 8 + T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGLb GaaiiEaiaacchakmaabmaabaacbmqcLbsacaWFibaakiaawIcacaGL Paaajugibiabg2da9iaa=LeakmaaBaaaleaajugWaiaaiIdaaSqaba qcLbsacqGHRaWkkmaaqahabaWaaSaaaeaajugibiaa=HeakmaaCaaa leqabaqcLbmacaWGRbaaaaGcbaqcLbsacaWGRbGaaiyiaaaaaSqaaK qzadGaam4Aaiabg2da9iaaigdaaSqaaKqzadGaeyOhIukajugibiab ggHiLdGaeyypa0Jaa8xsaOWaaSbaaSqaaKqzadGaaGioaaWcbeaaju gibiabgUcaRiaa=rfakmaaBaaaleaajugWaiaadggaaSqabaaaaa@5A20@                        (66) where I 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFjbGcdaWgaaWcbaqcLbmacaaI4aaaleqaaaaa@396C@ is an eight-order unit matrix. Using the addition theorem directly to add I 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFjbGcdaWgaaWcbaqcLbmacaaI4aaaleqaaaaa@396C@ and T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGHbaaleqaaaaa@399B@ when T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGHbaaleqaaaaa@399B@ is small relative to I 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFjbGcdaWgaaWcbaqcLbmacaaI4aaaleqaaaaa@396C@ , an error in the mantissa will occur due to computer rounding errors and leads to loss of precision.

Therefore, this paper uses an addition theorem to calculate T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGHbaaleqaaaaa@399B@

Φ 0 (Δξ)= [ ( I 8 + T a )( I 8 + T a ) ] 2 s1 = [ I 8 +2 T a + T a 2 ] 2 s1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccmqcLbsacq WFMoGrkmaaBaaaleaajugWaiaaicdaaSqabaqcLbsacaGGOaGaeuiL dqKaeqOVdGNaaiykaiabg2da9OWaamWaaeaadaqadaqaaGqadKqzGe Gaa4xsaOWaaSbaaSqaaKqzadGaaGioaaWcbeaajugibiabgUcaRiaa +rfakmaaBaaaleaajugWaiaadggaaSqabaaakiaawIcacaGLPaaada qadaqaaKqzGeGaa4xsaOWaaSbaaSqaaKqzadGaaGioaaWcbeaajugi biabgUcaRiaa+rfakmaaBaaaleaajugWaiaadggaaSqabaaakiaawI cacaGLPaaaaiaawUfacaGLDbaadaahaaWcbeqaaKqzadGaaGOmaSWa aWbaaWqabeaajugWaiaadohacqGHsislcaqGXaaaaaaajugibiabg2 da9OWaamWaaeaajugibiaa+LeakmaaBaaaleaajugWaiaaiIdaaSqa baqcLbsacqGHRaWkcaaIYaGaa4hvaOWaaSbaaSqaaKqzadGaamyyaa WcbeaajugibiabgUcaRiaa+rfakmaaDaaaleaajugWaiaadggaaSqa aKqzadGaaGOmaaaaaOGaay5waiaaw2faamaaCaaaleqabaqcLbmaca aIYaWcdaahaaadbeqaaKqzadGaam4CaiabgkHiTiaabgdaaaaaaaaa @74C7@     (67)

< T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGHbaaleqaaaaa@399B@ can be assumed as

T a =2 T a + T a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGHbaaleqaaKqzGeGaeyypa0JaaGOm aiaa=rfakmaaBaaaleaajugWaiaadggaaSqabaqcLbsacqGHRaWkca WFubGcdaqhaaWcbaqcLbmacaWGHbaaleaajugWaiaaikdaaaaaaa@459C@         (68)

After the N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36A9@ circulating assignment of Eq. (68), Eq. (67) can be written as

Φ 0 (Δξ)= e UΔξ = I 8 + T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccmqcLbsacq WFMoGrkmaaBaaaleaajugWaiaaicdaaSqabaqcLbsacaGGOaGaeuiL dqKaeqOVdGNaaiykaiabg2da9iaadwgakmaaCaaaleqabaacbmqcLb macaGFvbGaeuiLdqKaeqOVdGhaaKqzGeGaeyypa0Jaa4xsaOWaaSba aSqaaKqzadGaaGioaaWcbeaajugibiabgUcaRiaa+rfakmaaBaaale aajugWaiaadggaaSqabaaaaa@4FA8@                 (69)

Assuming segment coefficient matrix T k+1 = e UΔξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGRbGaey4kaSIaaGymaaWcbeaajugi biabg2da9iaadwgakmaaCaaaleqabaqcLbmacaWFvbGaeuiLdqKaeq OVdGhaaaaa@4325@ , the relationship of the state vector of each node can be described as

Z( ξ i+1 )= T i+1 Z( ξ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamyAaiabgUcaRiaa igdaaSqabaqcLbsacaGGPaGaeyypa0Jaa8hvaOWaaSbaaSqaaKqzad GaamyAaiabgUcaRiaaigdaaSqabaqcLbsacaWFAbGaaiikaiabe67a 4PWaaSbaaSqaaKqzadGaamyAaaWcbeaajugibiaacMcaaaa@4C38@      (70)

Z( ξ i+2 )= T i+2 Z( ξ i+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWHAb Gaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamyAaiabgUcaRiaaikda aSqabaqcLbsacaGGPaGaeyypa0dcbmGaa8hvaOWaaSbaaSqaaKqzad GaamyAaiabgUcaRiaaikdaaSqabaqcLbsacaWFAbGaaiikaiabe67a 4PWaaSbaaSqaaKqzadGaamyAaiabgUcaRiaaigdaaSqabaqcLbsaca GGPaaaaa@4DDF@     (71)

Z( ξ k+1 )= T k+1 Z( ξ k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaam4AaiabgUcaRiaa igdaaSqabaqcLbsacaGGPaGaeyypa0Jaa8hvaOWaaSbaaSqaaKqzad Gaam4AaiabgUcaRiaaigdaaSqabaqcLbsacaWFAbGaaiikaiabe67a 4PWaaSbaaSqaaKqzadGaam4AaaWcbeaajugibiaacMcaaaa@4C3E@     (72)

Z( ξ n )= T n Z( ξ n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamOBaaWcbeaajugi biaacMcacqGH9aqpcaWFubGcdaWgaaWcbaqcLbmacaWGUbaaleqaaK qzGeGaa8NwaiaacIcacqaH+oaEkmaaBaaaleaajugWaiaad6gacqGH sislcaaIXaaaleqaaKqzGeGaaiykaaaa@4AB5@        (73)

Solutions for the coefficient matrix of the conical shell

To facilitate the numerical calculation, the conical shell is split into a series of segments along the generatrix direction. Eq. (63) can be written as

Z( ξ 1 )=exp[ ξ 0 ξ 1 U(τ)dτ ]Z( ξ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGymaaWcbeaajugi biaacMcacqGH9aqpciGGLbGaaiiEaiaacchakmaadmaabaWaa8qmae aajugibiaa=vfacaGGOaGaeqiXdqNaaiykaiaadsgacqaHepaDaSqa aKqzadGaeqOVdG3cdaWgaaadbaqcLbmacaaIWaaameqaaaWcbaqcLb macqaH+oaEkmaaBaaameaajugibiaaigdaaWqabaaajugibiabgUIi YdaakiaawUfacaGLDbaajugibiaa=PfacaGGOaGaeqOVdGNcdaWgaa WcbaqcLbmacaaIWaaaleqaaKqzGeGaaiykaaaa@5DC8@     (74)

Z( ξ 2 )=exp[ ξ 1 ξ 2 U(τ)dτ ]Z( ξ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugi biaacMcacqGH9aqpciGGLbGaaiiEaiaacchakmaadmaabaWaa8qmae aajugibiaa=vfacaGGOaGaeqiXdqNaaiykaiaadsgacqaHepaDaSqa aKqzadGaeqOVdG3cdaWgaaadbaqcLbmacaaIXaaameqaaaWcbaqcLb macqaH+oaElmaaBaaameaajugWaiaaikdaaWqabaaajugibiabgUIi YdaakiaawUfacaGLDbaajugibiaa=PfacaGGOaGaeqOVdGNcdaWgaa WcbaqcLbmacaaIXaaaleqaaKqzGeGaaiykaaaa@5E6C@     (75)

Z( ξ j+1 )=exp[ ξ j ξ j+1 U(τ)dτ ]Z( ξ j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamOAaiabgUcaRiaa igdaaSqabaqcLbsacaGGPaGaeyypa0JaciyzaiaacIhacaGGWbGcda WadaqaamaapedabaqcLbsacaWFvbGaaiikaiabes8a0jaacMcacaWG KbGaeqiXdqhaleaajugWaiabe67a4TWaaSbaaWqaaKqzadGaamOAaa adbeaaaSqaaKqzadGaeqOVdG3cdaWgaaadbaqcLbmacaWGQbGaey4k aSIaaGymaaadbeaaaKqzGeGaey4kIipaaOGaay5waiaaw2faaKqzGe Gaa8NwaiaacIcacqaH+oaEkmaaBaaaleaajugWaiaadQgaaSqabaqc LbsacaGGPaaaaa@6274@     (76)

Z( ξ i )=exp[ ξ i1 ξ i U(τ)dτ ]Z( ξ i1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamyAaaWcbeaajugi biaacMcacqGH9aqpciGGLbGaaiiEaiaacchakmaadmaabaWaa8qmae aajugibiaa=vfacaGGOaGaeqiXdqNaaiykaiaadsgacqaHepaDaSqa aKqzadGaeqOVdG3cdaWgaaadbaqcLbmacaWGPbGaeyOeI0IaaGymaa adbeaaaSqaaKqzadGaeqOVdG3cdaWgaaadbaqcLbmacaWGPbaameqa aaqcLbsacqGHRiI8aaGccaGLBbGaayzxaaqcLbsacaWFAbGaaiikai abe67a4TWaaSbaaeaajugWaiaadMgacqGHsislcaaIXaaaleqaaKqz GeGaaiykaaaa@627C@     (77)

Assuming

T j+1 =exp[ ξ j ξ j+1 U(τ)dτ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGQbGaey4kaSIaaGymaaWcbeaajugi biabg2da9iGacwgacaGG4bGaaiiCaOWaamWaaeaadaWdXaqaaKqzGe Gaa8xvaiaacIcacqaHepaDcaGGPaGaamizaiabes8a0bWcbaqcLbma cqaH+oaElmaaBaaameaajugWaiaadQgaaWqabaaaleaajugWaiabe6 7a4TWaaSbaaWqaaKqzadGaamOAaiabgUcaRiaaigdaaWqabaaajugi biabgUIiYdaakiaawUfacaGLDbaaaaa@57DF@     (78)

Eqs. (75)-(78) can be described as

Z( ξ 1 )= T 1 Z( ξ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGymaaWcbeaajugi biaacMcacqGH9aqpcaWFubGcdaWgaaWcbaqcLbmacaaIXaaaleqaaK qzGeGaa8NwaiaacIcacqaH+oaEkmaaBaaaleaajugWaiaaicdaaSqa baqcLbsacaGGPaaaaa@4864@      (79)

Z( ξ 2 )= T 2 Z( ξ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWHAb Gaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiaa cMcacqGH9aqpieWacaWFubGcdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaa8NwaiaacIcacqaH+oaEkmaaBaaaleaajugWaiaaigdaaSqa baqcLbsacaGGPaaaaa@486F@    (80)

Z( ξ j+1 )= T j+1 Z( ξ j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamOAaiabgUcaRiaa igdaaSqabaqcLbsacaGGPaGaeyypa0Jaa8hvaOWaaSbaaSqaaKqzad GaamOAaiabgUcaRiaaigdaaSqabaqcLbsacaWFAbGaaiikaiabe67a 4PWaaSbaaSqaaKqzadGaamOAaaWcbeaajugibiaacMcaaaa@4C3B@     (81)

Z( ξ i )= T i Z( ξ i1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamyAaaWcbeaajugi biaacMcacqGH9aqpcaWFubGcdaWgaaWcbaqcLbmacaWGPbaaleqaaK qzGeGaa8NwaiaacIcacqaH+oaEkmaaBaaaleaajugWaiaadMgacqGH sislcaaIXaaaleqaaKqzGeGaaiykaaaa@4AA6@    (82)

The coefficient matrix U(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbsaca WFvbGaaiikaiabe67a4jaacMcaaaa@3A61@ for the conical shell is dependent on ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEaaa@3828@ . Therefore, the transfer matrix exp[ ξ j ξ j+1 U(τ)dτ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGLb GaaiiEaiaacchakmaadmaabaWaa8qmaeaaieWajugibiaa=vfacaGG OaGaeqiXdqNaaiykaiaadsgacqaHepaDaSqaaKqzadGaeqOVdG3cda WgaaadbaqcLbmacaWGQbaameqaaaWcbaqcLbmacqaH+oaElmaaBaaa meaajugWaiaadQgacqGHRaWkcaaIXaaameqaaaqcLbsacqGHRiI8aa GccaGLBbGaayzxaaaaaa@517A@ cannot be calculated like the transfer matrix for a cylindrical shell. This paper calculates the transfer matrix T j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGQbGaey4kaSIaaGymaaWcbeaaaaa@3B41@ for a conical shell by precise integration. Segments Δξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHuo arcqaH+oaEaaa@398F@ of the conical shell are divided into a precise integral step Δς MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHuo arcqaHcpGvaaa@3971@ (Δς= Δξ s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaeyiLdqKaeqOWdyLaeyypa0JcdaWcaaqaaKqzGeGaeyiLdqKaeqOV dGhakeaajugibiaadohaaaGaaiykaaaa@4134@ A value of 5 is recommended for s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiaeeaaaaa aaa8dapeGaa83Caaaa@3738@ . For the segment ξ j ξ j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEkmaaBaaaleaajugWaiaadQgaaSqabaqcLbsacqWI8iIocqaH+oaE kmaaBaaaleaajugWaiaadQgacqGHRaWkcaaIXaaaleqaaaaa@41FC@ of the conical shell, the integral step node is ς k = ξ j +k( ξ j+1 ξ j )/s= ξ j +kΔς MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHcp GvkmaaBaaaleaajugWaiaadUgaaSqabaqcLbsacqGH9aqpcqaH+oaE kmaaBaaaleaajugWaiaadQgaaSqabaqcLbsacqGHRaWkcaWGRbGcda qadaqaaKqzGeGaeqOVdGNcdaWgaaWcbaqcLbmacaWGQbGaey4kaSIa aGymaaWcbeaajugibiabgkHiTiabe67a4PWaaSbaaSqaaKqzadGaam OAaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaai4laiaadohacqGH9aqp cqaH+oaEkmaaBaaaleaajugWaiaadQgaaSqabaqcLbsacqGHRaWkca WGRbGaeuiLdqKaeqOWdyfaaa@5CD4@ , k=0,1,..,s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb Gaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaac6cacaGGUaGaaiil aiaadohaaaa@3E3C@ . In a precise integral step, assuming τ=( ς k1 + ς k )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDcqGH9aqpkmaabmaabaqcLbsacqaHcpGvkmaaBaaaleaajugWaiaa dUgacqGHsislcaaIXaaaleqaaKqzGeGaey4kaSIaeqOWdyLcdaWgaa WcbaqcLbmacaWGRbaaleqaaaGccaGLOaGaayzkaaqcLbsacaGGVaGa aGOmaaaa@487B@ , U(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFvbGaaiikaiabes8a0jaacMcaaaa@3A65@ can be recognized to be the constant coefficient matrix, which is independent of ς MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHcp Gvaaa@380A@ . The variable coefficient matrix T j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGQbGaey4kaSIaaGymaaWcbeaaaaa@3B41@ in segment Δξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHuo arcqaH+oaEaaa@398F@ of the conical shell can be calculated through the constant coefficient matrix of integral steps tiered multiplication

T j+1 = k=1 s exp[ U( τ k )( ς k ς k1 ) ] = k=1 s exp[ U( τ k )Δς ] = k=1 s T j+1 k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaWgaaWcbaqcLbmacaWGQbGaey4kaSIaaGymaaWcbeaajugi biaab2dakmaarahabaqcLbsaciGGLbGaaiiEaiaacchakmaadmaaba qcLbsacaWFvbGaaiikaiabes8a0PWaaSbaaSqaaKqzadGaam4AaaWc beaajugibiaacMcacaGGOaGaeqOWdyLcdaWgaaWcbaqcLbmacaWGRb aaleqaaKqzGeGaeyOeI0IaeqOWdyLcdaWgaaWcbaqcLbmacaWGRbGa eyOeI0IaaGymaaWcbeaajugibiaacMcaaOGaay5waiaaw2faaaWcba qcLbmacaWGRbGaeyypa0JaaGymaaWcbaqcLbmacaWGZbaajugibiab g+GivdGaeyypa0JcdaqeWbqaaKqzGeGaciyzaiaacIhacaGGWbGcda WadaqaaKqzGeGaa8xvaiaacIcacqaHepaDkmaaBaaaleaajugWaiaa dUgaaSqabaqcLbsacaGGPaGaeyiLdqKaeqOWdyfakiaawUfacaGLDb aaaSqaaKqzadGaam4Aaiabg2da9iaaigdaaSqaaKqzadGaam4Caaqc LbsacqGHpis1aiabg2da9OWaaebCaeaajugibiaa=rfakmaaDaaale aajugWaiaadQgacqGHRaWkcaaIXaaaleaajugWaiaadUgacqGHRaWk caaIXaaaaaWcbaqcLbmacaWGRbGaeyypa0JaaGymaaWcbaqcLbmaca WGZbaajugibiabg+Givdaaaa@8C00@      (83)

The constant coefficient matrix T j+1 k+1 =exp[ U( τ k )Δς ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFubGcdaqhaaWcbaqcLbmacaWGQbGaey4kaSIaaGymaaWcbaqcLbma caWGRbGaey4kaSIaaGymaaaajugibiabg2da9iGacwgacaGG4bGaai iCaOWaamWaaeaajugibiaa=vfacaGGOaGaeqiXdqNcdaWgaaWcbaqc LbmacaWGRbaaleqaaKqzGeGaaiykaiabgs5aejabek8awbGccaGLBb Gaayzxaaaaaa@4FF0@ of the precise integral step can be solved like the reference method for solving e UΔξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb GcdaahaaWcbeqaaGqadKqzadGaa8xvaiabfs5aejabe67a4baaaaa@3CBF@ in section 3.1.

Solutions for the point matrix at the junction of the coupled shell
The displacements, slopes, forces and moments at the junction of the coupled conical-cylindrical shell should satisfy the following deformation compatibility conditions (Figure 3) (Figure 4).
a. At the junction of the coupled shell, the displacements and slopes u co , v co , w co , φ co MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b GcdaWgaaWcbaqcLbmacaWGJbGaam4BaaWcbeaajugibabaaaaaaaaa peGaaiila8aacaWG2bGcdaWgaaWcbaqcLbmacaWGJbGaam4BaaWcbe aajugib8qacaGGSaWdaiaadEhakmaaBaaaleaajugWaiaadogacaWG VbaaleqaaKqzGeWdbiaacYcapaGaeqOXdOMcdaWgaaWcbaqcLbmaca WGJbGaam4BaaWcbeaaaaa@4C69@ of the conical shell are in continuity with u cy , v cy , w cy , φ cy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b GcdaWgaaWcbaqcLbmacaWGJbGaamyEaaWcbeaajugibabaaaaaaaaa peGaaiila8aacaWG2bGcdaWgaaWcbaqcLbmacaWGJbGaamyEaaWcbe aajugib8qacaGGSaWdaiaadEhakmaaBaaaleaajugWaiaadogacaWG 5baaleqaaKqzGeWdbiaacYcapaGaeqOXdOMcdaWgaaWcbaqcLbmaca WGJbGaamyEaaWcbeaaaaa@4C91@ of the cylindrical shell at the three coordinate axis.
b. At the junction of the coupled shell, the forces and moments N co s , S co sφ , N co s , M co s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaqhaaWcbaqcLbmacaWGJbGaam4BaaWcbaqcLbmacaWGZbaaaKqz Geaeaaaaaaaaa8qacaGGSaWdaiaadofakmaaDaaaleaajugWaiaado gacaWGVbaaleaajugWaiaadohacqaHgpGAaaqcLbsapeGaaiila8aa caWGobGcdaqhaaWcbaqcLbmacaWGJbGaam4BaaWcbaqcLbmacaWGZb aaaKqzGeWdbiaacYcapaGaamytaOWaa0baaSqaaKqzadGaam4yaiaa d+gaaSqaaKqzadGaam4Caaaaaaa@5564@ of the conical shell are equal to N cy s , S cy sφ , N cy s , M cy s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZbaaaKqz Geaeaaaaaaaaa8qacaGGSaWdaiaadofakmaaDaaaleaajugWaiaado gacaWG5baaleaajugWaiaadohacqaHgpGAaaqcLbsapeGaaiila8aa caWGobGcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZb aaaKqzGeWdbiaacYcapaGaamytaOWaa0baaSqaaKqzadGaam4yaiaa dMhaaSqaaKqzadGaam4Caaaaaaa@558C@ of the cylindrical shell at the three coordinate axis.

According to the positive directions shown in the Figure 3 and Figure 4, the displacements, slopes, forces and moments of the conical and cylindrical shells at the junction satisfy

u cy = u co cosα w co sinα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b GcdaWgaaWcbaqcLbmacaWGJbGaamyEaaWcbeaajugibiabg2da9iaa dwhakmaaBaaaleaajugWaiaadogacaWGVbaaleqaaKqzGeGaci4yai aac+gacaGGZbGaeqySdeMaeyOeI0Iaam4DaOWaaSbaaSqaaKqzadGa am4yaiaad+gaaSqabaqcLbsaciGGZbGaaiyAaiaac6gacqaHXoqyaa a@4FC9@    (84)

v cy = v co MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b GcdaWgaaWcbaqcLbmacaWGJbGaamyEaaWcbeaajugibiabg2da9iaa dAhakmaaBaaaleaajugWaiaadogacaWGVbaaleqaaaaa@4090@      (85)

w cy = u co sinα+ w co cosα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b GcdaWgaaWcbaqcLbmacaWGJbGaamyEaaWcbeaajugibiabg2da9iaa dwhakmaaBaaaleaajugWaiaadogacaWGVbaaleqaaKqzGeGaci4Cai aacMgacaGGUbGaeqySdeMaey4kaSIaam4DaOWaaSbaaSqaaKqzadGa am4yaiaad+gaaSqabaqcLbsaciGGJbGaai4BaiaacohacqaHXoqyaa a@4FC0@      (86)

φ cy = φ co MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAkmaaBaaaleaajugWaiaadogacaWG5baaleqaaKqzGeGaeyypa0Ja eqOXdOMcdaWgaaWcbaqcLbmacaWGJbGaam4BaaWcbeaaaaa@4214@        (87)

N cy s = N co s cosα+ V co s sinα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZbaaaKqz GeGaeyypa0JaamOtaOWaa0baaSqaaKqzadGaam4yaiaad+gaaSqaaK qzadGaam4CaaaajugibiGacogacaGGVbGaai4Caiabeg7aHjabgUca RiaadAfakmaaDaaaleaajugWaiaadogacaWGVbaaleaajugWaiaado haaaqcLbsaciGGZbGaaiyAaiaac6gacqaHXoqyaaa@55C4@      (88)

S cy sφ = S co sφ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZbGaeqOX dOgaaKqzGeGaeyypa0Jaam4uaOWaa0baaSqaaKqzadGaam4yaiaad+ gaaSqaaKqzadGaam4CaiabeA8aQbaaaaa@4812@      (89)

V cy s = N co s sinα+ V co s cosα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZbaaaKqz GeGaeyypa0JaeyOeI0IaamOtaOWaa0baaSqaaKqzadGaam4yaiaad+ gaaSqaaKqzadGaam4CaaaajugibiGacohacaGGPbGaaiOBaiabeg7a HjabgUcaRiaadAfakmaaDaaaleaajugWaiaadogacaWGVbaaleaaju gWaiaadohaaaqcLbsaciGGJbGaai4BaiaacohacqaHXoqyaaa@56B9@      (90)

M cy s = M co s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GcdaqhaaWcbaqcLbmacaWGJbGaamyEaaWcbaqcLbmacaWGZbaaaKqz GeGaeyypa0JaamytaOWaa0baaSqaaKqzadGaam4yaiaad+gaaSqaaK qzadGaam4Caaaaaaa@448C@          (91)

To take into consideration displacements, slopes, forces and moments satisfying the continuity conditions at the junction, the relationship of the left end state vector and the right end state vector at the junction is

Z(s= L cy L )= P cocy Z(s= L co R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcLbsaca WFAbGaaiikaGqaciaa+nhaieaacaqF9aGaa4htaOWaa0baaSqaaKqz adGaam4yaiaadMhaaSqaaKqzadGaamitaaaajugibiaacMcacaqF9a acbeGaaWhuaOWaaWbaaSqabeaajugWaiaadogacaWGVbGaeyOKH4Qa am4yaiaadMhaaaqcLbsacaWFAbGaaiikaiaa+nhacaqF9aGaa4htaO Waa0baaSqaaKqzadGaam4yaiaad+gaaSqaaKqzadGaamOuaaaajugi biaacMcaaaa@54DE@       (92)

Eq. (92) can be written as

Z( ξ i cy )= P cocy Z( ξ i co ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaqhaaWcbaqcLbmacaWGPbaaleaa jugWaiaadogacaWG5baaaKqzGeGaaiykaGqaaiaa+1daieqacaqFqb GcdaahaaWcbeqaaKqzadGaam4yaiaad+gacqGHsgIRcaWGJbGaamyE aaaajugibiaa=PfacaGGOaGaeqOVdGNcdaqhaaWcbaqcLbmacaWGPb aaleaajugWaiaadogacaWGVbaaaKqzGeGaaiykaaaa@554E@        (93)

The point transfer matrix P cocy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbsaca WFqbGcdaahaaWcbeqaaKqzadGaam4yaiaad+gacqGHsgIRcaWGJbGa amyEaaaaaaa@3E54@ can be described as

P cocy =[ cosα 0 sinα 0 0 0 0 0 0 1 0 0 0 0 0 0 sinα 0 cosα 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 cosα 0 sinα 0 0 0 0 0 0 1 0 0 0 0 0 0 sinα 0 cosα ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaju gibiaa=bfakmaaCaaaleqabaqcLbmacaWGJbGaam4BaiabgkziUkaa dogacaWG5baaaKqzGeGaeyypa0JcdaWadaqaaKqzGeqbaeqabGacaa aaaaaakeaajugibiGacogacaGGVbGaai4Caiabeg7aHbGcbaqcLbsa caaIWaaakeaajugibiabgkHiTiGacohacaGGPbGaaiOBaiabeg7aHb GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGc baqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcba qcLbsacaaIXaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqc LbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLb sacaaIWaaakeaajugibiGacohacaGGPbGaaiOBaiabeg7aHbGcbaqc LbsacaaIWaaakeaajugibiGacogacaGGVbGaai4Caiabeg7aHbGcba qcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqc LbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLb sacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGymaaGcbaqcLbsa caaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsaca aIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaI WaaakeaajugibiaaicdaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIWa aakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaa keaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaaicdaaOqaaKqzGeGaci4yaiaac+gacaGGZbGaeqySdega keaajugibiaaicdaaOqaaKqzGeGaeyOeI0Iaci4CaiaacMgacaGGUb GaeqySdegakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsa caaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsaca aIWaaakeaajugibiaaigdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaI WaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWa aakeaajugibiaaicdaaOqaaKqzGeGaci4CaiaacMgacaGGUbGaeqyS degakeaajugibiaaicdaaOqaaKqzGeGaci4yaiaac+gacaGGZbGaeq ySdegaaaGccaGLBbGaayzxaaaaaa@B8FD@      (94)

Solutions for the coefficient matrix of the coupled shell
An illustration of the coupled conical-cylindrical shell is seen in Figure 5, α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyaaa@3824@ where is the semi-vertex conical angle. R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb aaaa@375C@ is the radius of the cylindrical shell, which is also the larger end radius of the conical shell. L s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A41@ is the length from the top point of the conical shell to the smaller end of conical shell along the generatrix direction. L e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaSbaaSqaaKqzadGaamyzaaWcbeaaaaa@3A33@ is the length from the top point of the conical shell to the larger end of the conical shell along the generatrix direction. The length of conical shell is L co = L e L s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb GcdaWgaaWcbaqcLbmacaWGJbGaam4BaaWcbeaajugibiabg2da9iaa dYeakmaaBaaaleaajugWaiaadwgaaSqabaqcLbsacqGHsislcaWGmb GcdaWgaaWcbaqcLbmacaWGZbaaleqaaaaa@43F4@ and the length of the cylindrical shell is L cy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaSbaaSqaaKqzadGaam4yaiaadMhaaSqabaaaaa@3B2F@ . The thickness of the coupled shell is h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb aaaa@3772@ .

According to sections 3.1, 3.2, and 3.3, the state vector of the segment nodes from the coupled conical-cylindrical shell satisfy

Z( ξ 1 )= T 1 Z( ξ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWcbaqcLbmacaaIXaaaleqa aKqzGeGaaiykaiabg2da9iaa=rfakmaaBaaaleaajugWaiaaigdaaS qabaqcLbsacaWFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGim aaWcbeaajugibiaacMcaaaa@4A19@      (95)

Z( ξ 2 )= T 2 Z( ξ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWcbaqcLbmacaaIYaaaleqa aKqzGeGaaiykaiabg2da9iaa=rfakmaaBaaaleaajugWaiaaikdaaS qabaqcLbsacaWFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGym aaWcbeaajugibiaacMcaaaa@4A1C@     (96)

Z( ξ i co )= T i Z( ξ i1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaqhaaWcbaqcLbmacaWGPbaaleaa jugWaiaadogacaWGVbaaaKqzGeGaaiykaiabg2da9iaa=rfakmaaBa aaleaajugWaiaadMgaaSqabaqcLbsacaWFAbGaaiikaiabe67a4PWa aSbaaSqaaKqzadGaamyAaiabgkHiTiaaigdaaSqabaqcLbsacaGGPa aaaa@4F66@     (97)

Z( ξ i cy )= P cocy Z( ξ i co ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaqhaaWcbaqcLbmacaWGPbaaleaa jugWaiaadogacaWG5baaaKqzGeGaaiykaGqaaiaa+1daieqacaqFqb GcdaahaaWcbeqaaKqzadGaam4yaiaad+gacqGHsgIRcaWGJbGaamyE aaaajugibiaa=PfacaGGOaGaeqOVdGNcdaqhaaWcbaqcLbmacaWGPb aaleaajugWaiaadogacaWGVbaaaKqzGeGaaiykaaaa@554E@    (98)

Z( ξ i+1 )= T i+1 Z( ξ i cy )= T i+1 P cocy Z( ξ i co ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWcbaqcLbmacaWGPbGaey4k aSIaaGymaaWcbeaajugibiaacMcacqGH9aqpcaWFubGcdaWgaaWcba qcLbmacaWGPbGaey4kaSIaaGymaaWcbeaajugibiaa=PfacaGGOaGa eqOVdGNcdaqhaaWcbaqcLbmacaWGPbaaleaajugWaiaadogacaWG5b aaaKqzGeGaaiykaiabg2da9iaa=rfakmaaBaaaleaajugWaiaadMga cqGHRaWkcaaIXaaaleqaaGqabKqzGeGaa4huaOWaaWbaaSqabeaaju gWaiaadogacaWGVbGaeyOKH4Qaam4yaiaadMhaaaqcLbsacaWFAbGa aiikaiabe67a4PWaa0baaSqaaKqzadGaamyAaaWcbaqcLbmacaWGJb Gaam4BaaaajugibiaacMcaaaa@69D1@     (99)

Z( ξ n )= T n Z( ξ n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaju gibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWcbaqcLbmacaWGUbaaleqa aKqzGeGaaiykaiabg2da9iaa=rfakmaaBaaaleaajugWaiaad6gaaS qabaqcLbsacaWFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaamOB aiabgkHiTiaaigdaaSqabaqcLbsacaGGPaaaaa@4C6A@    (100)

Eqs. (95)-(100) can be written in term of a matrix as follows

[ T 1 I 0 0 0 0 0 0 0 T 2 I 0 0 0 0 0 0 0 T 3 I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T i+1 P I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T n I ] ( 8n,8n+8 ) { Z( ξ 0 ) Z( ξ 1 ) Z( ξ 2 ) Z( ξ i ) Z( ξ n ) } ( 8n+8,1 ) ={ 0 0 0 0 0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWada qaaKqzGeqbaeqabCacaaaaaaGcbaqcLbsacqGHsislieWacaWFubGc daWgaaWcbaqcLbmacaaIXaaaleqaaaGcbaqcLbsacaWFjbaakeaaju gibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugi biaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibi aaicdaaOqaaKqzGeGaeyOeI0Iaa8hvaOWaaSbaaSqaaKqzadGaaGOm aaWcbeaaaOqaaKqzGeGaa8xsaaGcbaqcLbsacaaIWaaakeaajugibi aaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaa icdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiabgk HiTiaa=rfakmaaBaaaleaajugWaiaaiodaaSqabaaakeaajugibiaa =LeaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaic daaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicda aOqaaKqzGeGaaGimaaGcbaqcLbsacaGGUaGaaiOlaiaac6caaOqaaK qzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqz GeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiabgkHiTiaa=rfakmaa BaaaleaajugWaiaadMgacqGHRaWkcaaIXaaaleqaaGqabKqzGeGaa4 huaaGcbaqcLbsacaWFjbaakeaajugibiaaicdaaOqaaKqzGeGaaGim aaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaa GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaiOlaiaa c6cacaGGUaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLb sacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsa caaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacq GHsislcaWFubGcdaWgaaWcbaqcLbmacaWGUbaaleqaaaGcbaqcLbsa caWFjbaaaaGccaGLBbGaayzxaaWaaSbaaSqaaOWaaeWaaSqaaKqzad GaaGioaiaad6gacaGGSaGaaGioaiaad6gacqGHRaWkcaaI4aaaliaa wIcacaGLPaaaaeqaaOWaaiWaaeaajugibuaabeqaheaaaaGcbaqcLb sacaWFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGaaGimaaWcbeaa jugibiaacMcaaOqaaKqzGeGaa8NwaiaacIcacqaH+oaEkmaaBaaale aajugWaiaaigdaaSqabaqcLbsacaGGPaaakeaajugibiaa=PfacaGG OaGaeqOVdGNcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaaiykaa GcbaqcLbsacqWIUlstaOqaaKqzGeGaa8NwaiaacIcacqaH+oaEkmaa BaaaleaajugWaiaadMgaaSqabaqcLbsacaGGPaaakeaajugibiabl6 UinbGcbaqcLbsacaWFAbGaaiikaiabe67a4PWaaSbaaSqaaKqzadGa aeOBaaWcbeaajugibiaacMcaaaaakiaawUhacaGL9baadaWgaaWcba GcdaqadaWcbaqcLbmacaaI4aGaamOBaiabgUcaRiaaiIdacaGGSaGa aGymaaWccaGLOaGaayzkaaaabeaajugibiabg2da9OWaaiWaaeaaju gibuaabeqaheaaaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqa aKqzGeGaaGimaaGcbaqcLbsacqWIUlstaOqaaKqzGeGaaGimaaGcba qcLbsacqWIUlstaOqaaKqzGeGaaGimaaaaaOGaay5Eaiaaw2haaaaa @E893@      (101)

According to the given boundary conditions at the ends of the coupled shell, row numbers where elements of the state vector are zero are found. Then, to delete corresponding columns of coefficient matrix, Eq. (101) can be written as

[ T 1 I 0 0 0 0 0 0 0 T 2 I 0 0 0 0 0 0 0 T 3 I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T i+1 P I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T n I ] ( 8n,8n ) { Z ( ξ 0 ) Z( ξ 1 ) Z( ξ 2 ) Z( ξ i ) Z ( ξ n ) } ( 8n,1 ) = { 0 0 0 0 0 } ( 8n,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWada qaaKqzGeqbaeqabCacaaaaaaGcbaqcLbsacqGHsislkmaaIaaabaac bmqcLbsacaWFubGcdaWgaaWcbaqcLbmacaaIXaaaleqaaaGccaGLIm caaeaajugibiaa=LeaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaa keaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacqGHsislcaWF ubGcdaWgaaWcbaqcLbmacaaIYaaaleqaaaGcbaqcLbsacaWFjbaake aajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaa jugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaaju gibiaaicdaaOqaaKqzGeGaeyOeI0Iaa8hvaOWaaSbaaSqaaKqzadGa aG4maaWcbeaaaOqaaKqzGeGaa8xsaaGcbaqcLbsacaaIWaaakeaaju gibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugi biaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibi aac6cacaGGUaGaaiOlaaGcbaqcLbsacaaIWaaakeaajugibiaaicda aOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaO qaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqa aKqzGeGaeyOeI0Iaa8hvaOWaaSbaaSqaaKqzadGaamyAaiabgUcaRi aaigdaaSqabaacbeqcLbsacaGFqbaakeaajugibiaa=LeaaOqaaKqz GeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGa aGimaaGcbaqcLbsacaGGUaGaaiOlaiaac6caaOqaaKqzGeGaaGimaa GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGc baqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcba qcLbsacaaIWaaakeaajugibiabgkHiTiaa=rfakmaaBaaaleaajugW aiaad6gaaSqabaaakeaadaGiaaqaaKqzGeGaa8xsaaGccaGLImcaaa aacaGLBbGaayzxaaWaaSbaaSqaaOWaaeWaaSqaaKqzadGaaGioaiaa d6gacaGGSaGaaGioaiaad6gaaSGaayjkaiaawMcaaaqabaGcdaGada qaaKqzGeqbaeqabCqaaaaakeaadaGiaaqaaKqzGeGaa8NwaaGccaGL ImcajugibiaacIcacqaH+oaEkmaaBaaaleaajugWaiaaicdaaSqaba qcLbsacaGGPaaakeaajugibiaa=PfacaGGOaGaeqOVdGNcdaWgaaWc baqcLbmacaaIXaaaleqaaKqzGeGaaiykaaGcbaqcLbsacaWFAbGaai ikaiabe67a4PWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiaacMca aOqaaKqzGeGaeSO7I0eakeaajugibiaa=PfacaGGOaGaeqOVdGNcda WgaaWcbaqcLbmacaWGPbaaleqaaKqzGeGaaiykaaGcbaqcLbsacqWI UlstaOqaamaaIaaabaqcLbsacaWFAbaakiaawkYiaKqzGeGaaiikai abe67a4PWaaSbaaSqaaKqzadGaaeOBaaWcbeaajugibiaacMcaaaaa kiaawUhacaGL9baadaWgaaWcbaGcdaqadaWcbaqcLbmacaaI4aGaam OBaiaacYcacaaIXaaaliaawIcacaGLPaaaaeqaaKqzGeGaeyypa0Jc daGadaqaaKqzGeqbaeqabCqaaaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiabl6UinbGcbaqcLbsa caaIWaaakeaajugibiabl6UinbGcbaqcLbsacaaIWaaaaaGccaGL7b GaayzFaaWaaSbaaSqaaOWaaeWaaSqaaKqzadGaaGioaiaad6gacaGG SaGaaGymaaWccaGLOaGaayzkaaaabeaaaaa@F145@     (102)

Since the state vectors cannot all be zero vectors, the determinant of the coefficient matrix must be zero. The following equation is obtained.

| T 1 I 0 0 0 0 0 0 0 T 2 I 0 0 0 0 0 0 0 T 3 I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T i+1 P I 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 T n I | ( 8n,8n ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabda qaaKqzGeqbaeqabCacaaaaaaGcbaqcLbsacqGHsislkmaaIaaabaac bmqcLbsacaWFubGcdaWgaaWcbaqcLbmacaaIXaaaleqaaaGccaGLIm caaeaajugibiaa=LeaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaa keaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacqGHsislcaWF ubGcdaWgaaWcbaqcLbmacaaIYaaaleqaaaGcbaqcLbsacaWFjbaake aajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaa jugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaaju gibiaaicdaaOqaaKqzGeGaeyOeI0Iaa8hvaOWaaSbaaSqaaKqzadGa aG4maaWcbeaaaOqaaKqzGeGaa8xsaaGcbaqcLbsacaaIWaaakeaaju gibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugi biaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibi aac6cacaGGUaGaaiOlaaGcbaqcLbsacaaIWaaakeaajugibiaaicda aOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaO qaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqa aKqzGeGaeyOeI0Iaa8hvaOWaaSbaaSqaaKqzadGaamyAaiabgUcaRi aaigdaaSqabaacbeqcLbsacaGFqbaakeaajugibiaa=LeaaOqaaKqz GeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGa aGimaaGcbaqcLbsacaGGUaGaaiOlaiaac6caaOqaaKqzGeGaaGimaa GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGc baqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcba qcLbsacaaIWaaakeaajugibiabgkHiTiaa=rfakmaaBaaaleaajugW aiaad6gaaSqabaaakeaadaGiaaqaaKqzGeGaa8xsaaGccaGLImcaaa aacaGLhWUaayjcSdWaaSbaaSqaaOWaaeWaaSqaaKqzadGaaGioaiaa d6gacaGGSaGaaGioaiaad6gaaSGaayjkaiaawMcaaaqabaqcLbsacq GH9aqpcaaIWaaaaa@A84A@     (103)

The natural frequencyof the coupled conical-cylindrical shell is the only unknown quantity in the matrix T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHub aaaa@3868@ and is obtained through solving the frequency characteristic Eq. (103). By substituting the natural frequency into Eq. (102), the proportional relationship of state vectors can be obtained. Then, the modes of the coupled shell will be acquired in the given boundary condition.

Figure 3 The positive directions for displacements and slopes of conical and cylindrical shells.
Figure 4 The positive directions for forces and moments of conical and cylindrical shells.
Figure 5 Illustration for the coupled conical-cylindrical shell.

Conclusion

A new method, PITMM, is introduced in this paper to research the free vibrational characteristics of isotropic coupled conical-cylindrical shells. Based on the traditional transfer matrix and precise integration methods, the PITMM is constructed. The method not only retains the traditional transfer matrix methods’ advantages of formula regularity and easy programming but also obtains the high accuracy from the precise integration methods.

Acknowledgement

The authors gratefully acknowledge the financial support from the National Natural Science Foundation China (No.51209052).

Conflicts of Interest

None.

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