Dynamics of a rotating ring-stiffened sandwich conical shell with an auxetic honeycomb core

The free vibration analysis of a rotating sandwich conical shell with a reentrant auxetic honeycomb core and homogenous isotropic face layers reinforced with a ring support is studied.The shell is modeled utilizing the first-order shear deformation theory(FSDT)incorporating the relative,centripetal,and Coriolis accelerations alongside the initial hoop tension created by the rotation.The governing equations,compatibility conditions,and boundary conditions are attained using Hamilton’s principle.Utilizing trigonometric functions,an analytical solution is derived in the circumferential direction,and a numerical one is presented in the meridional direction via the differential quadrature method(DQM).The effects of various factors on the critical rotational speeds and forward and backward frequencies of the shell are studied.The present work is the first theoretical work regarding the dynamic analysis of a rotating sandwich conical shell with an auxetic honeycomb core strengthened with a ring support.

Torsional postbuckling characteristics of functionally graded graphene enhanced laminated truncated conical shell with temperature dependent material properties

Buckling and postbuckling characteristics of laminated graphene-enhanced composite(GEC)truncated conical shells exposed to torsion under temperature conditions using finite element method(FEM)simulation are presented in this study.In the thickness direction,the GEC layers of the conical shell are ordered in a piece-wise arrangement of functionally graded(FG)distribution,with each layer containing a variable volume fraction for graphene reinforcement.To calculate the properties of temperaturedependent material of GEC layers,the extended Halpin-Tsai micromechanical framework is used.The FEM model is verified via comparing the current results obtained with the theoretical estimates for homogeneous,laminated cylindrical,and conical shells,the FEM model is validated.The computational results show that a piece-wise FG graphene volume fraction distribution can improve the torque of critical buckling and torsional postbuckling strength.Also,the geometric parameters have a critical impact on the stability of the conical shell.However,a temperature rise can reduce the crucial torsional buckling torque as well as the GEC laminated truncated conical shell’s postbuckling strength.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

THE CONSTRUCTION OF WAVELET-BASED TRUNCATED CONICAL SHELL ELEMENT USING B-SPLINE WAVELET ON THE INTERVAL

Based on B-spline wavelet on the interval （BSWI）, two classes of truncated conical shell elements were constructed to solve axisymmetric problems, i.e. BSWI thin truncated conical shell element and BSWI moderately thick truncated conical shell element with independent slopedeformation interpolation. In the construction of wavelet-based element, instead of traditional polynomial interpolation, the scaling functions of BSWI were employed to form the shape functions through the constructed elemental transformation matrix, and then construct BSWI element via the variational principle. Unlike the process of direct wavelets adding in the wavelet Galerkin method, the elemental displacement field represented by the coefficients of wavelets expansion was transformed into edges and internal modes via the constructed transformation matrix. BSWI element combines the accuracy of B-spline function approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples of conical shells were studied to demonstrate the present element with higher efficiency and precision than the traditional element.

Buckling analysis of shear deformable composite conical shells reinforced by CNTs subjected to combined loading on the two-parameter elastic foundation

The main objective of this study is to investigate the buckling analysis of CCSs reinforced by CNTs subjected to combined loading of hydrostatic pressure and axial compression resting on the twoparameter elastic foundation(T-P-EF).It is one of the first attempts to derive the governing equations of the CCSs reinforced with CNTs,based on a generalized first-order shear deformation shell theory(FSDST)which includes shell-foundation interaction.By adopting the extended mixing rule,the effective material properties of CCSs reinforced by CNTs with linear distributions are approximated by introducing some efficiency parameters.Three carbon nanotube distribution in the matrix,i.e.uniform distribution(U)and V and X-types linear distribution are taken into account.The stability equations are solved by using the Galerkin procedure to determine the combined buckling loads(CBLs)of the structure selected here.The numerical illustrations cover CBLs characteristics of CCSs reinforced by CNTs in the presence of the T-P-EF.Finally,a parametric study is carried out to study the influences of the foundation parameters,the volume fraction of carbon nanotubes and the types of reinforcement on the CBLs.

Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

The generalized differential quadrature method （GDQM） is employed to con- sider the free vibration and critical speed of moderately thick rotating laminated compos- ite conical shells with different boundary conditions developed from the first-order shear deformation theory （FSDT）. The equations of motion are obtained applying Hamilton＇s concept, which contain the influence of the centrifugal force, the Coriolis acceleration, and the preliminary hoop stress. In addition, the axial load is applied to the conical shell as a ratio of the global critical buckling load. The governing partial differential equations are given in the expressions of five components of displacement related to the points ly- ing on the reference surface of the shell. Afterward, the governing differential equations are converted into a group of algebraic equations by using the GDQM. The outcomes are achieved considering the effects of stacking sequences, thickness of the shell, rotating velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes indicate that the rate of the convergence of frequencies is swift, and the numerical tech- nique is superior stable. Three comparisons between the selected outcomes and those of other research are accomplished, and excellent agreement is achieved.

Buckling analysis of functionally graded material(FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material （FGM） coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable- coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

Influence of elastic foundations and carbon nanotube reinforcement on the hydrostatic buckling pressure of truncated conical shells

In this study,the effects of elastic foundations(EFs)and carbon nanotube(CNT)reinforcement on the hydrostatic buckling pressure(HBP)of truncated conical shells(TCSs)are investigated.The first order shear deformation theory(FOSDT)is generalized to the buckling problem of TCSs reinforced with CNTs resting on the EFs for the first time.The material properties of composite TCSs reinforced with CNTs are graded linearly according to the thickness coordinate.The Winkler elastic foundation(W-EF)and Pasternak elastic foundation(P-EF)are considered as the EF.The basic relations and equations of TCSs reinforced with CNTs on the EFs are obtained in the framework of the FOSDT and solved using the Galerkin method.One of the innovations in this study is to obtain a closed-form solution for the HBP of TCSs reinforced with CNTs on the EFs.Finally,the effects of the EFs and various types CNT reinforcements on the HBP are investigated simultaneously.The obtained results are compared with the results in the literature,and the accuracy of results is confirmed.

Modeling and optimal vibration control of conical shell with piezoelectric actuators

In this paper numerical simulations of active vibration control for conical shell structure with dis-tributed piezoelectric actuators is presented.The dynamic equations of conical shell structure are derivedusing the finite element model (FEM) based on Mindlin's plate theory.The results of modal calculationswith FEM model are accurate enough for engineering applications in comparison with experiment results.The Electromechanical influence of distributed piezoelectric actuators is treated as a boundary conditionfor estimating the control force.The independent modal space control (IMSC) method is adopted and theoptimal linear quadratic state feedback control is implemented so that the best control performance withthe least control cost can be achieved.Optimal control effects are compared with controlled responses withother non-optimal control parameters.Numerical simulation results are given to demonstrate the effective-ness of the control scheme.

THERMOELASTICALLY COUPLED AXISYMMETRIC NONLINEAR VIBRATION OF SHALLOW SPHERICAL AND CONICAL SHELLS

The problem of axisymmetric nonlinear vibration for shallow thin spherical and conical shells when temperature and strain fields are coupled is studied. Based on the large deflection theories of von Krmn and the theory of thermoelasticity, the whole governing equations and their simplified type are derived. The time-spatial variables are separated by Galerkin's technique, thus reducing the governing equations to a system of time-dependent nonlinear ordinary differential equation. By means of regular perturbation method and multiple-scales method, the first-order approximate analytical solution for characteristic relation of frequency vs amplitude parameters along with the decay rate of amplitude are obtained, and the effects of different geometric parameters and coupling factors as well as boundary conditions on thermoelastically coupled nonlinear vibration behaviors are discussed.

A NEW DISPLACEMENT-TYPE STABILITY EQUATION AND GENERAL STABILITY ANALYSIS OF LAMINATED COMPOSITE CIRCULAR CONICAL SHELLS WITH TRIANGULAR GRID STIFFENERS

In this paper, based on the theory of Donnell-type shallow shell, a new displacement-type stability equations is first developed for laminated composite circular conical shells with triangular grid stiffeners by using the variational calculus and generalized smeared-stiffener theory. The most general bending stretching couplings, the effect of eccentricity of stiffeners are considered. Then, for general stability of composite triangular grid stiffened conical shells without twist coupling terms, the approximate formulas are obtained for critical external pressure by using Galerkin's procedure. Numerical examples for a certain C/E composite conical shells with inside triangular grid stiffeners are calculated and the results are in good agreement with the experimental data. Finally, the influence of some parameters on critical external pressure is studied. The stability equations developed and the formulas for critical external pressure obtained in this paper should be very useful in the astronautical engineering design.

A DONNELL TYPE THEORY FOR FINITE DEFLECTION OF STIFFENED THIN CONICAL SHELLS COMPOSED OF COMPOSITE MATERIALS

A Donnell type theory is developed for finite deflection of closely stiffened truncated laminated composite conical shells under arbitrary loads by using the variational calculus and smeared-stiffener theory. The most general bending-stretching coupling and the effect of eccentricity of stiffeners are considered. The equilibrium equations, boundary conditions and the equation of compatibility are derived. The new equations of the mixed-type of stiffened laminated composite conical shells are obtained in terms of the transverse deflection and stress function. The simplified equations are also given for some commonly encountered cases.

THE DISPLACEMENT SOLUTION OF CONICAL SHELL AND ITS APPLICATION

In this paper, the displacement solution method of the conical shell is presented. From the differential equations in displacement form of conical shell and by introducing a displacement function, U,the differential equations are changed into an eight-order soluble partial differential equation about the displacement Junction U in which the coefficients are variable. A t the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function. As special cases of this paper, the displacement function introduced by V. Z. Vlasov in circular cylindrical shell, the basic equation of the cylindrical shell and that of the circular plate are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell is reduced to finding the displacement functionU,and the general solution of the governing equation is obtained in generalized hypergeometric function, For the axisymmetric bending deformation of the conical shell, the general solution is expressed in the Bessel functionOn the basis of the governing equation obtained in this paper, the differential equation of conical shell on the elastic foundation (A Winkler Medium) is deduced, its general solutions are given in a power series, and the numerical calculations are carried out.

NON-SYMMETRICAL LARGE DEFORMATION OFA SHALLOW THIN CONICAL SHELL

In this paper, non-symmetrical large deformation problem of a shallow conical shell is studied by two-parameter perturbation method. The third-order approximate analytical solution of the deformation of a shallow conical shell subjected to linear loads is obtained and the characteristic curves of load-deflection on a perturbing point are portrayed. The similar questions of other kind of shell and plate can be discussed by using this paper's method. As the examples, the large deflection of plate and shallow conical shells with different initial deflections is discussed.

Structural and acoustic response of a finite stiffened conical shell

In this paper, a precise transfer matrix method is presented to calculate the struc- tural and acoustic responses of the conical shell. The governing equations of conical shells are written as a coupled set of first order differential equations. The field transfer matrix of the shell and non-homogenous term resulting from the external excitation are obtained by precise integra- tion method. After assembling the field transfer matrixes, the whole matrix describing dynamic behavior of the stiffened conical shell is obtained. Then the structural and acoustic responses of the shell are solved by obtaining unknown sound pressure coefficients. The natural frequencies of the shell are compared with the FEM results to test the validity. Furthermore, the effects of the semi-vertex angle, driving force directions and boundary conditions on the structural and acoustic responses are studied.

Couple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers

The size-dependent geometrically nonlinear harmonically soft excited oscillation of composite truncated conical microshells(CTCMs)made of functionally graded materials(FGMs)integrated with magnetostrictive layers is analyzed.It is supposed that the FGM CTCMs are subjected to mechanical soft excitations together with external magnetic fields.An analytical framework is created by a microstructuredependent shell model having the 3rd-order distribution of shear deformation based on the modified couple stress(MCS)continuum elasticity.With the aid of the discretized form of differential operators developed via the generalized differential quadrature technique,a numerical solution methodology is introduced for obtaining the couple stress-based amplitude and frequency responses related to the primary resonant dynamics of the FGM CTCMs.Jump phenomena due to the loss of the first stability branch and falling down to the lower stable branch can be seen in the nonlinear primary resonance of the FGM CTCMs.It is demonstrated that the hardening type of nonlinearity results in bending the frequency response to the right side,and the MCS type of size effect weakens this pattern.Moreover,for higher material gradient indexes,the hardening type of nonlinearity is enhanced,and the MCS-based frequency response bends more considerably to the right side.

Buckling Analysis of FG Porous Truncated Conical Shells Resting on Elastic Foundations in the Framework of the Shear Deformation Theory

In this article,an analytical method is proposed to analyze of the linear buckling behavior of the FG porous truncated conical shells subjected to a uniform axial compressive load and resting on the Pasternak elastic foundation.The material properties including Young’s modulus,shear modulus and density are assumed to vary in the thickness direction.Three types of FG porous distributions including symmetric porosity distribution,non-symmetric porosity and uniform porosity distribution are considered.The governing equations of the FG porous truncated conical shells are obtained by using the first-order shear deformation theory(FSDT).With the help of the Galerkin method,the expressions for critical buckling loads are obtained in closed forms.The reliability of the obtained results is verified by comparing the present solutions with the published solutions.Finally,the numerical results show the effects of shell characteristics,porosity distribution,porosity coefficient,and elastic foundation on the critical buckling load.

NONLINEAR STABILITY OF TRUNCATED SHALLOW CONICAL SANDWICH SHELL WITH VARIABLE THICKNESS

The theory of nonlinear stability for a truncated shallow conical shell with variable thickness under the action of uniform pressure was presented. The fundamental equations and boundary conditions were derived by means of calculus of variations. An analytic solution for the critical buckling pressure of the shell with a hyperbolically varying thickness is obtained by use of modified iteration method. The results of numerical calculations are presented in diagrams, which show the influence of geometrical and physical parameters on the buckling behavior.

NONLINEAR VIBRATIONS OF ORTHOTROPIC SHALLOW SHELLS OF REVOLUTION

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.

NONLINEAR FREE VIBRATION OF ORTHOTROPIC SHALLOW SHELLS OF REVOLUTION UNDER THE STATIC LOADS

In this paper, the axisymmetric nonlinear free vibration problems of cylindrically orthotropic shallow thin spherical and conical shells under uniformly distributed static loads are studied by using MWR and Lindstedt-Poincare perturbation method, from which, the characteristic relation between frequency ratio and amplitude is obtained. The effects of static loads, geometric and material parameters on vibrational behavior of shells are also discussed.