Shell Theory Research Articles

Simulation and Optimization of Hemispherical Resonator's Equivalent Bottom Angle for Frequency-Splitting Suppression.

As an inertial sensor with excellent performance, the hemispherical resonator gyro is widely used in aerospace, weapon navigation and other fields due to its advantages of high precision, high reliability, and long life. Due to the uneven distributions of material properties and mass of the resonator in the circumferential direction, the frequencies of the two 4-antinodes vibration modes (operational mode) of resonator in different directions are different, which is called frequency splitting. Frequency splitting is the main error source affecting the accuracy of the hemispherical resonator gyro and must be suppressed. The frequency splitting is related to the structure of the resonator. For the planar-electrode-type hemispherical resonator gyro, in order to suppress the frequency splitting from the structure, improve the accuracy of the hemispherical resonator gyro, and determine and optimize the equivalent bottom angle parameters of the hemispherical resonator, this paper starts from the thin shell theory, and the 4-antinodes vibration mode and waveform precession model of the hemispherical resonator are researched. The effect of the equivalent bottom angle on the 4-antinodes vibration mode frequency value under different boundary conditions is theoretically analyzed and simulated. The simulation results show that the equivalent bottom angle affects the 4-antinodes vibration mode of the hemispherical resonator through radial constraints. The hemispherical resonator with mid-surface radius R=15 mm and shell thickness h=1 mm is the optimization object, and the stem diameter D and fillet radius R1 are experimental factors, with the 4-antinodes vibration mode frequency value and mass sensitivity factor as the response indexes. The central composite design is carried out to optimize the equivalent bottom angle parameters. The optimized structural parameters are: stem diameter D=7 mm, fillet radii R1=1 mm, R2=0.8 mm. The simulation results show that the 4-antinodes vibration mode frequency value is 5441.761 Hz, and the mass sensitivity factor is 3.91 Hz/mg, which meets the working and excitation requirements wonderfully. This research will provide guidance and reference for improving the accuracy of the hemispherical resonator gyro.

Observation and Thermodynamic Analysis of Reversible Fluid Isobars in Isolated Single Digit Nanopore Carbon Nanotubes

Recent interest in one-dimensionally confined fluids, where confinement approaches molecular dimensions, has demonstrated exceptionally high fluxes from slip flow and large distortions of fluid phase boundaries [1,2]. The Center for Enhanced Nanofluidic Transport (CENT) was recently formed as an intellectual hub for studying extreme fluid confinement in what we label Single Digit Nanopores (SDNs). To this end, in this work, we note that predicting such phenomena for a given conduit dimension has been confounded by a dearth of fundamental thermodynamic measurements and analysis as a function of confinement diameter and wall composition. Herein, we utilize Raman spectroscopy and carbon nanotubes with diameters less than 3 nm to identify and study newly observed, thermally driven, fluid adsorbed phases under conditions at constant ambient pressure, tracing what we label a fluid isobar as a function of temperature. We show that the Raman radial breathing(-like) vibrational modes (RBLMs) can serve as fluid sensors to trace the dependence of the fluid isobar and allow study of basic thermodynamics of fluid phase transition as a function of conduit curvature and confinement volume. We study fluids on the exterior and interior of double- and single- walled carbon nanotubes spanning 0.9 to 3.3 nm in diameter, showing that the fluid location can be disambiguated on double walled tubes by using an adaptation of elastic shell theory that allows deconvolution of inner and outer shell frequencies with fluid location. We find that the RBLM FWHM systematically changes as a unique signature of the location of the fluid phase near the CNT wall. A Buckingham potential is used along with electron diffraction assigned DWNT to provide a high accuracy estimate of the carbon-carbon coupling constant for two elastic shells, an important structural parameter for DWNT. The thermodynamic analysis of isobars measured for more than 25 CNTs under various environmental conditions is conducted by comparison to a new fluid equation of state that provides a continuum description of the fluid isobar under confinement. Our work sets forth a new and reliable approach for studying phase change of different fluids in large numbers of isolated single digit nanopores that may inform the refinement of fluid force fields and motivate theoretical refinements to the calculation of thermodynamic properties under confinement.[1] S. Faucher, N. Aluru, M. Z. Bazant, D. Blankschtein, A. H. Brozena, J. Cumings, J. Pedro De Souza, M. Elimelech, R. Epsztein, J. T. Fourkas, A. G. Rajan, H. J. Kulik, A. Levy, A. Majumdar, C. Martin, M. McEldrew, R. P. Misra, A. Noy, T. A. Pham, M. Reed, E. Schwegler, Z. Siwy, Y. Wang, M. S. Strano. Critical Knowledge Gaps in Mass Transport through Single-Digit Nanopores: A Review and Perspective. J. Phys. Chem. C 123, 21309–21326 (2019).[2] K. V. Agrawal, S. Shimizu, L. W. Drahushuk, D. Kilcoyne, M. S. Strano. Observation of extreme phase transition temperatures of water confined inside isolated carbon nanotubes. Nat. Nanotechnol. 12, 267–273 (2017)

On dynamic of imperfect GNP nanocomposite joined hemisphere-cylinder shells on Winkler foundation

This paper aims to analyze the frequencies related to the poriferous nanocomposite joined hemisphere-cylinder shells (JHCS) resting on the Winkler foundation for the first time. In detail, the Winkler foundation is used to model the soil’s behavior surrounding the JHCS. Graphene nanoplatelet (GNP) is a nano-size material that enhances a matrix to make GNP nanocomposites (GNPNs). Additionally, nine different functionally graded (FG) distribution forms along the thickness of the JHCS are used to mark the efficiency of GNP through the GNPN. In addition, two distribution models along the thickness of the JHCS, covering FG and uniform models, are implemented to address the effect of the porosity along the GNPN. Supplementarily, via joining Donnell’s shell and first shear deformation theories, the primary equations of the JHCS are determined. Addedly, Hamilton’s principle finds the fundamental motion equations (FMEs) of the JHCS. The generalized differential quadrature scheme discretizes the FMEs, boundary, and joining equations. Next, via eigenvalue determination, the frequencies of the system are discovered. Finally, the influences correlated with physical values, FG forms of the GNP, edge cases, porosity models, and Winkler foundation on the frequencies correlated with the poriferous nanocomposite JHCS are evaluated.

Investigation on Free Vibration of Rotating Cylindrical Shells with Variable Thickness

In order to improve the performance and efficiency of the rotating cylindrical shell (RCS), one of the effective ways reduces the mass of the RCS. The scientific and effective method is to design the variable thickness of RCS (VTRCS) along the axial direction in response to this demand. Then, the vibration traveling wave characteristics of VTRCS are investigated by Chebyshev–Ritz method. The displacement field of the cylindrical shell is expanded in the form of the product of the Chebyshev polynomial and the boundary function. The kinetic and potential energies of the VTRCS are calculated based on the Sanders shell theory considering the effects of Coriolis forces and centrifugal forces. Also, the frequency equation of the VTRCS is obtained according to the Ritz method. The accuracy of the modeling method is verified by comparing the present results with literature that had done, and the convergence of the calculated results is studied. Finally, the free-vibration traveling wave characteristics of the VTRCS in different thickness variations are compared, and the parameters such as the rotational speed, thickness variation parameters, and aspect ratio on the free-vibration traveling wave characteristics of the VTRCS are discussed. The influence of thickness variation on the vibration characteristics is analyzed, which has significance for the lightweight design for VTRCS.

Nonlinear Oscillations and Buckling Prediction for Shallow Convex Shells

Since 2010, the testing of thin-walled shallow shells has revealed that the ultimate internal stresses, under which the shells lose their stability (collapse), appeared to be lower than those predicted by the buckling theory used in American and European standards. The existing norms are based on the static theory of shallow shells proposed in 1930 with stable solutions for thin-walled structures within the nonlinear theory. These stable solutions differ significantly from the forms of equilibrium common to small initial loads. The minimum load, under which an alternative form of equilibrium exists, was used as the ultimate load. In the 1970s, this approach was proved to be unacceptable for complex loadings that were not a part of the real world in the past but are now the case with thinner products operated under complex conditions. Therefore, the initial theories on bearing capacity evaluations must be revisited. The new theory could be built on the basis of recent mathematical results establishing that the estimates made per two schemes (three-dimensional dynamic theory of elasticity and dynamic theory of shallow convex shells) were asymptotically close. Green's special function is introduced to convert the Foppl-von Karman system into a resolving integro-differential equation. The obtained nonlinear equation allows for the separation of variables and has numerous time-periodic solutions that satisfy the Duffing equation with a "soft spring". Numerical analysis enables determining the amplitude and period of oscillations depending on the properties of Green's function. This paper reviews an experimental setup in which oscillations are generated with the probing load directed normally to the surface of the shell. The experimental measurements of the resonance displacements as a function of coordinates and time enable the calculation of the safety factor for the bearing capacity of the structure using the non-destructive method under operating conditions.

Free Vibration Analyses of Stiffened Functionally Graded Graphene-Reinforced Composite Multilayer Cylindrical Panel

In this paper, the free vibration response of a stiffened functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer cylindrical shell panel is studied for the first time. The shell is stiffened by both stringers and rings. Additionally, the effect of reinforcing the shell panel, ring and stinger with GPLs is independently studied. Halpin–Tsai relations are employed to evaluate the mechanical properties of the shell panel, rings and stringers. The first-order shear deformation shell theory, accompanied by the Lekhnitsky smeared stiffener model, using the numerical finite element method and Hamilton principle, is employed to develop the governing motion equations of the shell panel. Four different types of GPL patterns, including FG-A, FG-X, FG-O and UD, are assumed across the thickness of the shell panel, rings and stringers. The effects of different factors, including various weight fractions and patterns of GPLs nanofillers, the geometry of the shell panel and stiffeners and two displacement boundary conditions, on the natural frequencies of the shell panel, have been studied.

On the Static Instability of FG-GNP-Reinforced Composite Cylindrical Shells Under Thermo-Mechanical Loading

The nonlinear buckling response of laminated composite cylindrical shells reinforced with graphene nanoplatelets (GNPs) is studied in this paper. The functionally graded (FG) shell reinforced by GNPs is analytically studied under external pressure and uniform temperature rise loadings. It is also assumed that the GNP-reinforced laminated composite shell is in contact with an elastic foundation. Various types of profiles are employed for the GNP distribution patterns in the shell thickness including 10 nanocomposite layers. The nonlinear strain-displacement relations of the shallow cylindrical panel are established utilizing the third-order shear deformation shell theory. Governing equilibrium equations of the laminated GNP-reinforced composite shell are formulated employing the principle of virtual displacement. The coupled system of nonlinear differential equations is solved analytically for the hinged–hinged and fixed–fixed boundaries of the shell using a perturbation-based technique. Correctness of presented formulations and obtained solutions is proved by comparisons with results from previous studies for an isotropic cylindrical shell. Novel numerical results reveal that the material properties, geometrical characteristics and load parameters significantly affect on the buckling behavior of laminated composite cylindrical shells.

Theoretical modeling of the pre-cutting system performance for the tunnel face stability in very weak rock masses

By forming continuous pre-lining around the tunnel's perimeter, pre-cutting is a recognized tunneling technique that enables full-face excavation of a tunnel even in unfavorable ground conditions. In this study, a theoretical method is put out to simulate the performance of this kind of pre-support in squeezing conditions. The loading distribution on the pre-cutting shells is determined for this purpose using the LDP of an unsupported tunnel, and the non-linear stiffness of the pre-cutting shell is then determined using the shell theory. The convergence confinement method is used to calculate how much the pre-supported tunnel isdeformed. The non-uniform loading distribution also results in the tangential and longitudinal bending moments on the pre-cutting shell. To show the effectiveness of pre-cutting shells in extremesqueezingconditions, several examples are solved. As a result, a few graphs for the preliminarydesign of a tunnel using this technique are given. The graphs show the overall tunnel displacement for various pre-cutting shells under various geological conditions. The outcomes demonstrate good agreement between FALC 3D resultsand the ones by thesuggested analytical model. A preliminary design can be made using the assessment of the proposed methodonthe loads on the pre-cutting system.

Nonlinear dynamics of bolted joined conical-cylindrical shells considering displacement-dependent characteristics

This paper investigates the nonlinear dynamic characteristics of joined conical-cylindrical shells (JCCSs) with bolt connection for the first time. In the process of developing mechanical model of the bolt connection, the displacement-dependent stiffness and damping are taken into account, which could simulate the change of the contact state at the connection interface under different excitations. Both theoretical and experimental studies are performed to illustrate the nonlinear vibration behaviors of the bolted JCCSs. Successive uniform distributed artificial springs between the conical and cylindrical shells are adopted to simulate the rigid joint of the two shells. Donnell's shell theory is employed in theoretical modeling, and the governing equation is obtained by the Lagrange equation. The displacement admissible function applied in the research is Chebyshev polynomial. By comparison with existing literature and experimental data, the accuracy of present theoretical model is validated. It is found that both theory and experiment show a dynamic softening phenomenon with the increase of excitation level. Taking the displacement-dependent characteristics in bolt connection zone into consideration can effectively illustrate this soft phenomenon. The established model is capable of predicting the nonlinear vibration characteristics of JCCSs with bolt connection.

Buckling Analysis of Grid-stiffened Composite Cylindrical Shell

An energy based smeared stiffener model is developed to obtain equivalent stiffness coefficients of a general grid-stiffened composite cylindrical shell. These equivalent stiffness coefficients are used in the Ritz buckling analysis of the shell. Transverse shear properties are important in a stiffened shell and results are obtained for First order Shear Deformation Theory (FSDT) as well as for Classical Laminated Shell Theory (CLST). Parametric study is carried out to find out the effects of various design parameters on specific buckling load of the shell.

Structural optimization of an encased bi-material FGP-GNF pipeline-liner system with novel polyhedral configurations

This article proposes a novel thin-walled polyhedral functionally graded porous (FGP) liner consolidated by graphene nanofillers (GNF) to rehabilitate the cracked pipeline. A buckle propagation may occur since the liner is subjected to hydrostatic pressure and temperature variational field simultaneously. A cosine function is developed to describe the deformed shape of the liner. Both pores and the GNF are distributed symmetrically on the cross-section of the liner, which also has a polyhedral profile to improve its bending stiffness. The nonlinear equilibrium equations are accurately obtained by introducing the thin-walled shell theory and the principle of minimum potential energy. The critical buckling pressure is obtained by solving the nonlinear equilibrium equations. Analytical solutions are compared successfully with other closed-form results to accomplish the verification when the FGP-GNF polyhedral liner reduces to an isotropic circular one. Furthermore, one improvement factor is proposed to analyze the effect of different polyhedral shapes on the buckling pressure of the encased polyhedral FGP-GNF liner. Finally, parameters are evaluated to further understand the buckling behavior of the FGP-GNF polyhedral liner, including the number of polygons, porosity coefficient, mass fraction, geometric shape of the GNF, radius-to-thickness ratio, temperature field, etc.

Vibration Control of Composite Thick Shells Using Magnetostrictive Material and Higher Order Shear Deformation Theory

Analytical solutions of laminated composite shells with embedded actuating layers are presented in this study. The actuating layers are used to control natural vibration of laminated composite shell panels. The first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) for shells are used to represent the shell kinematics and equations of motion. The exact solution for symmetric laminated shells with embedded actuating layers under simply supported boundary conditions is obtained using the Navier solution procedure. Negative velocity feedback control is used. The parametric effect of the position of the magnetostrictive layers, material properties and control parameters on the vibration suppression are investigated in detail. It is found that (i) the shortest vibration suppression time is achieved by placing the actuating layers farthest from the neutral axis (ii) using thinner smart material layers leads to better vibration attenuation characteristics and, (iii) the vibration suppression time is longer for a lower value of the feedback control coefficient. For thicker shells HSDT predicts larger amplitude of vibration and longer vibration suppression time as compared to FSDT predictions.

Research on prediction of critical pressure of FGM cylindrical shells under hydrostatic pressure based on homogenization transformation method

This study uses the classical Flügge shell theory and homogenization transformation method to predict the critical pressure of functionally graded material (FGM) cylindrical shells under hydrostatic pressure. The method considers the similarity in physical properties and mechanical behavior between FGM and uniform materials. The vibration equation of the coupled system of an underwater FGM cylindrical shells is derived using the wave method, taking into account the influence of fluids. The natural frequency of the FGM cylindrical shell under hydrostatic pressure is obtained using the Newton iteration method. Based on the linear correlation between critical load and load level with zero natural frequency, the fitting curve method and the formula method after homogenization transformation are used to predict and analyze the critical pressure of FGM cylindrical shells under hydrostatic pressure. The effects of various parameters on the critical pressure are discussed. The results show that the material elastic modulus E, geometric dimensions h/R and L/R of FGM cylindrical shells, and different boundary conditions have a significant impact on the critical pressure. Through multiple comparative analyses of examples, the correctness and effectiveness of the proposed method are verified. The method has high prediction accuracy and low computational cost, providing a new exploration path for the analysis of non-uniform structural mechanical behavior.

Nonlinear dynamic response and vibration of spiral stiffened FG toroidal shell segments with variable thickness

In this study, the nonlinear vibratory responses and primary resonance (PR) of the spiral stiffened functionally graded (SSFG)-toroidal shell segments (TSSs) are investigated. The shell segments considered are with variable thickness and subjected to external excitations. Both the concave and convex TSSs are investigated in this study. The TSSs considered are reinforced with internal spiral stiffeners and are assumed to be continuously graded along their thickness direction. To model the stiffeners, the smeared stiffeners technique is utilized. Also, the improved Donnell shell theory, von-Kármán nonlinear geometric assumptions, and Galerkin’s method are applied to derive the discretized nonlinear governing equation of SSFG-TSSs with variable thickness. To examine the PR responses, the multiple scales method is utilized. To investigate the nonlinear vibratory behaviors, the 4th order P-T method is used. Although the results of the present research are verified with previous research available in the literature, the response of nonlinear vibration is compared with the 4th order Runge–Kutta (R-K) method as well. The influences of geometrical and material parameters on the natural frequencies, nonlinear vibration, and PR responses of SSFG-TSSs are also studied.

Investigation of sound transmission through a finite length cylindrical shell in the presence of an exterior turbulent boundary layer

This paper studies sound transmission through a thin cylindrical shell of finite length. The structure is subjected to an exterior turbulent boundary layer (TBL) excitation and the boundary conditions of its two ends are considered as simply-supported. The classical shell theory (CST) is employed to derive wave propagation in the cylindrical shell, and the Corcos model is applied to describe pressure fluctuation due to TBL. According to the existence of two axial and radial modes, a complete convergence study is presented. To examine sound transmission through the structure, a comprehensive parameters study is provided on the resulting power spectral density (PSD) of the kinetic energy. To validate the results of the present method, the statistical energy analysis (SEA) method is used. For this purpose, modeling is performed by SEA in VA One software, which shows the high accuracy of the results of the present method. The analytical predictions suggest that the sound insulation performance of such a finite length structure is enhanced considerably by increasing the thickness of the shell in a broad-band frequency range. Similar effects on sound transmission are observed for Young’s modulus and density of the shell particularly in the low and high-frequency ranges, respectively. Whereas, enhancing the turbulent pressure and freestream velocity adversely affects sound radiation into the structure.

Vibration and thermomechanical transient response of doubly curved FGM panels with porosities and elastically restrained edges

The present paper aims to analyze the influences of porosities, elevated temperature and elasticity of in-plane boundary conditions on the free vibration and nonlinear transient response of functionally graded material (FGM) doubly curved panels subjected to uniform lateral pressure. Porosities are assumed to exist in the FGM according to even and uneven distributions. Effective properties of porous FGM are determined using a modified rule of mixture. Basic equations in terms of deflection and stress function are derived basing on the classical shell theory taking into account von Kármán–Donnell nonlinearity and initial geometrical imperfection. Analytical solutions for simply supported panels are assumed and Galerkin method along with Runge–Kutta scheme are applied to obtain the deflection–time response for nonlinear dynamic analysis. Numerical results demonstrate that tangential constraints of edges have extremely significant influences on the vibration and transient response of porous FGM panels exposed to an elevated temperature. The analysis also reveals that porosities have negative and positive effects on the dynamic response of FGM panels at reference and elevated temperatures, respectively. Furthermore, when the edges are restrained, initial imperfection importantly influence the behavior tendency of porous FGM panels exposed to preexist thermal environment.

Lagrange multipliers at the boundary in the inextensional bending theory of thin elastic shells

The theory of inextensional deformations of shells is extended by introducing Lagrange multipliers at the boundary to supplement those operating in the interior. These are associated with a constraint on the metric on the closure of the shell midsurface. The additional multipliers allow for a richer set of external actions than those admitted by the conventional theory. In particular, they balance concentrated forces in the tangent plane of the surface applied at corners of the boundary.

Robust optimization of prismatic lithium-ion cells for reducing thermal performance fluctuations and manufacturing costs

The thermal performances of lithium-ion cells are significantly influenced by unavoidable uncertainties during manufacturing. However, there is a lack of uncertainty optimization methods to control the thermal performance fluctuations of lithium-ion cells. This paper proposes a robust optimization approach for prismatic lithium-ion cells, aiming to optimize the manufacturing cost and control the thermal performance fluctuations without removing the uncertainties. The cost objective function and thermal robustness constraints are established considering the uncertainties of the manufacturing parameters, thereby formulating the robust optimization model of the prismatic lithium-ion cell. The robust optimization performs a nested optimization by optimizing the parameter standard deviations for an optimal cost and evaluating the cell thermal robustness at each design point. An analytical expression for the cell expansion behaviour is created based on the elastic shell theory. An electrochemical-thermal-mechanics multidisciplinary analysis method with ordinary differential equations is given to evaluate the cell thermal performances such as temperature, pressure, and expansion under thermal abuse. It addresses the efficiency issue in the nested optimization. The robust optimization approach is applied to design problems for two-type lithium-ion cells packaged in pouch and hard-pack. Numerical and experimental results demonstrate the excellent potential of the approach due to its advantages in applicability, accuracy, and efficiency.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.

Vibrational development of nanocomposite cylindrical shells by employing reduced graphene oxide (rGO) as a nanoscale strengthener

This study uses the harmonic differential quadrature technique to learn more about the free vibration of cylinder constructions made from reduced graphene oxide nanocomposite (rGON). To support, using the Halpin-Tsai homogenization form, the effective Young's modulus related to the rGON is considered here. In addition, five distinct functionally graded (FG) distribution types of reduced graphene oxide (rGO) are examined along shell thickness. The main equations of the cylinder structures are identified by combining the first-order shear deformation theory (FSDT), the general shell theory, and Hamilton's principle. A comprehensive analysis of the impact of changing factors, such as rGO weight percent and FG distribution types, is presented. With the significant results: Regarding the rGON cylinder structure reinforced by uniformly rGO, adding 0.1%, 0.5%, 1%, and 2% rGO to the polymer matrix can cause to improving the minimum natural frequency to 1.91%, 9.23%, 17.75%, and 33.17% improvements compare with a cylindrical shell made by the pure polymer.