Elastic Cylindrical Shell Research Articles

Elastic Response of Acoustic Coating on Fluid-Loaded Rib-Stiffened Cylindrical Shells

Reinforced cylindrical shells are used in numerous industries; common examples include undersea vehicles, aircraft, and industrial piping. Current models typically incorporate approximation theories to determine shell behavior, which are limited by both thickness and frequency. In addition, many applications feature coatings on the shell interior or exterior that normally have thicknesses which must also be considered. To increase the fidelity of such systems, this work develops an analytic model of an elastic cylindrical shell featuring periodically spaced ring stiffeners with a coating applied to the outer surface. There is an external fluid environment. Beginning with the equations of elasticity for a solid, spatial-domain displacement field solutions are developed incorporating unknown wave propagation coefficients. These fields are used to determine stresses at the boundaries of the shell and coating, which are then coupled with stresses from the stiffeners and fluid. The stress boundary conditions contain double-index infinite summations, which are decoupled, truncated, and recombined into a global matrix equation. The solution to this global equation results in the displacement responses of the system as well as the exterior scattered pressure field. An incident acoustic wave excitation is considered. Thin-shell reference models are used for validation, and the predicted system response to an example simulation is examined. It is shown that the reinforcing ribs and coating add significant complexity to the overall cylindrical shell model; however, the proposed approach enables the study of structural and acoustic responses of the coupled system.

Imperfection sensitivity in cylindrical shells under uniform bending

Efforts are ongoing to characterise a comprehensive resistance function for cylindrical shells under uniform bending, a ubiquitous structural system that finds application in load-bearing circular hollow sections, tubes, piles, pipelines, wind turbine support towers, chimneys and silos. A recent computational study by Rotter et al. demonstrated that nonlinear buckling of perfect elastic cylinders under bending is governed by four length-dependent domains –‘short’, ‘medium’, ‘transitional’ and ‘long’– depending on the relative influence of end boundary conditions and cross-sectional ovalisation. The study additionally transformed its resistance predictions into compact algebraic relationships for use as design equations within the recently developed framework of reference resistance design. This article extends on the above to present a detailed computational investigation into the imperfection sensitivity of thin elastic cylindrical shells across the most important length domains, using automation to carry out the vast number of necessary finite element analyses. Geometric imperfections in three forms – the classical linear buckling eigenmode, an imposed cross-sectional ovalisation and a realistic manufacturing ‘weld depression’ defect – are applied to demonstrate that imperfection sensitivity is strongly length dependent but significantly less severe than for the closely related load case of cylinders under uniform axial compression. The axisymmetric weld depression almost always controls as the most deleterious imperfection. The data are processed computationally to offer an accurate yet conservative lower-bound algebraic design characterisation of imperfection sensitivity for use within the RRD framework. The outcomes are relevant to researchers and designers of large metal shells under bending and will appeal to computational enthusiasts who are encouraged to adopt the automation methodology described herein to explore other structural systems.

The Study of Natural Oscillations of Thin Elastic Shells of Complicated Shapes by the Energy Method

The problem of natural oscillations of an elastic thin non-circular cylindrical wavy shell of an open profile is considered. The problem is based on the Rayleigh-Ritz energy method. On the basis of the proposed method for determining the lowest frequencies and forms of natural vibrations of shells of complicated shape, the numerical convergence of the developed algorithm is investigated. The evaluation of the results of this numerical experiment is given.

Numerical analysis on target strength of the air filled cylindrical elastic shell

Aimed at simulating the air filled finite cylindrical shell target strength, the numerical calculation method of scattering form function and target strength is given in this paper using the theory of Stanton T K. Cylindrical shells of different material properties, thickness, length are calculated, and the results indicate that factor of the material property and dimension is very important. Compared with other kinds of material properties, the gum rubber cylindrical shell has much more greater target strength for low frequency. Factors of radius and thickness have big effect to its fundamental resonance and the choice should be made in accordance with the actual needs. The numerical study provides theoretical basis and design reference for engineering application.

Active noise cloaking of 2D cylindrical shells

Active noise cloaking of rigid and elastic 2D cylindrical shells in an infinite acoustic medium is presented. The control approach utilises control sources or forces and discrete error sensors at which the desired field is specified. The optimal complex control source amplitudes are determined from minimisation of a quadratic cost function and a priori knowledge of the incident sound field. For the case of the rigid cylinder, the scattered field due to an incident plane wave is actively attenuated using monopole sources and error sensors arranged circumferentially around the scatterer. For the case of the elastic cylindrical shell, the scattered and radiated fields are actively attenuated at the shell circumferential modes using either monopole control sources in the exterior fluid or point control forces applied directly to excite the shell. Active cloaking using control forces to modify the structural response, herein termed active structural acoustic cloaking, is found to result in simultaneous cloaking of both the interior and exterior fields, attributed to direct coupling between the vibrating shell and the interior and exterior fluids. An approach to identify the number of control sources/forces and error sensors for active cloaking within a predefined error is also described.

Hydroelasticity of three elastic coaxial shells interacting with viscous incompressible fluids between them under vibration

Elastic cylindrical shells interacting with a viscous incompressible fluid are widely used in various branches of science and technology, such as engineering and aviation engineering. They provide the possibility for solving a lot of problems, such as: reducing constructing weight and dimensions, equalizing dynamic influences and vibration level, as well as reducing friction and wearing, cooling. Mathematical model of the system, representing three coaxial cylindrical shells, freely supported at the ends, and interacting with viscous incompressible fluid between them under mechanical system harmonic vibration is constructed. This mathematical model represents a coupled system consisting of the Navier-Stokes, continuity for each fluid and equations and the ones of elastic coaxial cylindrical shells dynamics which are based on the Kirchhoff-Love hypotheses and the corresponding boundary conditions, namely: for the fluid non-flow and for free attaching for the shells. The constructed mathematical model allows to investigate the oscillations of a mechanical system consisting of coaxial elastic cylindrical shells interacting with viscous incompressible liquids in order to identify dangerous operating modes.

Mathematical Modeling of Nonlinear Waves in an Elastic Cylindrical Shell Surrounded by an Elastic Medium and Containing a Viscous Incompressible Liquid

A mathematical model is proposed for a Kirchhoff–Love-type nonlinear elastic cylindrical shell surrounded by an elastic medium and containing a viscous incompressible liquid. The model is used to analyze wave processes both analytically and numerically. On the basis of the proposed computational algorithm, a software package is developed, which makes it possible to plot diagrams and to obtain numerical solutions to Cauchy problems with initial conditions taken in the form of exact solutions to dynamic equations of shells in the absence of the liquid.

Contact interaction of thin-walled elements with an elastic layer and an infinite circular cylinder under torsion

We consider a class of contact torsion problems on interaction of thin-walled elements shaped as an elastic thin washer – a flat circular plate of small height – with an elastic layer, in particular, with a half-space, and on interaction of thin cylindrical shells with a solid elastic cylinder, infinite in both directions. The governing equations of the physical models of elastic thin washers and thin circular cylindrical shells under torsion are derived from the exact equations of mathematical theory of elasticity using the Hankel and Fourier transforms. Within the framework of the accepted physical models, the solution of the contact problem between an elastic washer and an elastic layer is reduced to solving the Fredholm integral equation of the first kind with a kernel representable as a sum of the Weber–Sonin integral and some integral regular kernel, while solving the contact problem between a cylindrical shell and solid cylinder is reduced to a singular integral equation (SIE). An effective method for solving the governing integral equations of these problems are specified.

Exact solutions of smart laminated orthotropic cylindrical shells with interface diffusion and sliding

ABSTRACTIn this article, an exact solution is obtained to the time-dependent problem of a laminated orthotropic elastic cylindrical shell bonded to piezoelectric actuator and sensor incorporating interfacial diffusion and sliding subjected to electromechanical loadings. The state-space method is employed to solve this problem. Both the direct and the inverse piezoelectric effects of the piezoelectric material are investigated. The relaxation times and the transient electroelastic fields of displacements, stresses, and electric potential are calculated for three-layered and five-layered smart cylindrical shells. It is found that interfacial diffusion and sliding can effectively reduce the initial high stress concentrations as the evolution time is long enough. The derived solution can be used as benchmark results for various approximate shell theories and numerical methods.

Acoustic gradient index lens design using gradient based optimization

We demonstrate a gradient based optimization of the absolute acoustic pressure at a focal point by incrementally repositioning a set of cylindrical obstacles so that they eventually act as an effective gradient index (GRIN) lens. The gradient-based optimization algorithm maximizes the absolute pressure at the focal point by analytically evaluating its derivative with respect to the cylinder positions and then perturbatively optimizing the position of each cylinder in the GRIN lens device while taking into account acoustic multiple scattering between the cylinders. The method is presented by examples of sets of hard cylinders and sets of elastic cylindrical shells of uniform size. Designs using available cylindrical shells enable easy and realistic fabrication of acoustic GRIN lenses. Computations are performed on MATLAB using parallel optimization algorithms and a multistart optimization solver, and supplying the gradients of absolute pressure at the focal point with respect to position vectors. Providing the analytic form of the gradients enhances modeling by allowing the use of exact gradients with optimization algorithms and parallel computing. This results in reducing number of function calls and time needed to converge, and improving solution accuracy for large scale optimization problems especially at high frequencies and with a large number of scatterers.

Shell buckling, without ‘imperfections’

The buckling behaviour of thin shell structures under load has been a persistent challenge to engineering designers and researchers over many decades. In this article I consider two unusual experimental studies on the buckling of thin-walled elastic cylindrical shells, each of which sheds intriguing light on the buckling phenomena. The classical theory of buckling of thin cylindrical shells under axial compression predicts that the buckling stress will be proportional to t/ R– the ratio of thickness to radius – other things being equal. But collected results of experimental studies from many laboratories, when plotted on log–log scales, show clearly that the buckling stress is actually proportional to ( t/ R)1.5, with the measured buckling stresses being scattered through a factor of about 4 for shells with R/ t > 200. Such scatter is commonly judged to be in accord with Koiter’s theory of ‘imperfection sensitivity’. But that theory lays no claim to an understanding of the empirical 1.5-power law. I claim that a key to this situation is the experimental performance of some small-scale open-topped silicone rubber shells, buckling under their own weight, which clearly demonstrates a 1.5-power law, but with very little scatter. The buckling mode of these shells involves almost entirely inextensional deformation, with a single small dimple growing near the base, separated from the rest of the shell by a narrow boundary layer that accounts for almost all of the dimple’s elastic strain energy. A straightforward, simple analysis of the mechanics of the dimple is consistent with the experimental 1.5-power law. As noted above, experimental buckling loads of shells that are closed at both ends also show the empirical 1.5-power law, but now with significant statistical scatter. A second aim of the paper is to throw light on that phenomenon. I venture to attribute it to the effect of the boundary conditions of the shell. I adduce support for this view from experimental observations on the buckling of a shell with special, frictional end-fittings. That feature produces significantly higher collapse loads, and with much smaller scatter, than for corresponding shells with fixed boundaries; and it permits striking pre-buckled deformations to occur, of a kind not previously noted. It will be appreciated that neither of the two parts of this article depends on the widely accepted theory of imperfection-sensitivity; hence my choice of title. It is a pleasure for me to submit this article to a special publication in honour of Michael Rotter, with whom I have discussed matters of this sort through three decades.

Nonlinear behaviour of short elastic cylindrical shells under global bending

A recent computational study identified four distinct domains of stability behaviour at different lengths in thin elastic cylindrical shells under global bending. Cylinders of sufficient length suffer from fully-developed cross-sectional ovalisation and fail by local buckling at a moment very close to the Brazier prediction. Progressively shorter cylinders experience less ovalisation owing to the increasingly strong restraint provided by the boundary at the edges. Very short thin cylinders, however, restrain the formation of even a local buckle and fail through a limit point instability at moments and curvatures significantly in excess of the classical elastic prediction. This limit point behaviour is not caused by ovalisation but by the growth of a destabilising fold on the compressed meridian.The nonlinear behaviour of very short cylinders under global bending is investigated in detail herein, covering a wide range of lengths, radius to thickness ratios and boundary conditions with both restrained and unrestrained meridional rotations corresponding to ‘clamped’ and ‘simply-supported’ conditions respectively. Two types of imperfections are investigated, the critical buckling eigenmode and a realistic manufacturing-related ‘weld depression’. A complex insensitivity to these imperfections is revealed owing to a pre-buckling stress state dominated by local compatibility bending, and the cylinder length is confirmed as playing a crucial role in governing this behaviour. The study contributes to the characterisation of multi-segment shells with very short individual cylindrical segments, often found in the aerospace and marine industries as well as in specialised civil engineering applications such as LIPP® silos.

New Insight into Microvascular Perfusion by Analyzing Pipe Flow in a Cylindrical Elastic Shell

Background: Microvascular perfusion, a kind of regional perfusion, plays important roles in delivering oxygen and nutrients, and regulating blood pressure and responses to inflammation. Aim: The aim of this research is to analyze the characteristics of microvascular perfusion by conducting pipe flow in a circular elastic tube. Methods: A model was established with circular elastic tube to mimic microvascular perfusion. The velocity of pressure waves was calculated according to the time that the liquid took to spilt over. What’s more, the characteristics and significance of microvascular flow and arteriovenous anastomoses (AVAs) were analyzed. Results: It took the liquid about 0.1 second to spill over from the model, and the velocity of pressure waves is greater than 100 m/s in the elastic pipe. A mechanical switch structure and the corresponding mechanism were proposed for microvascular perfusion in AVAs. Conclusion: Microvascular perfusion maintains a considerable constancy of hemodynamics in different tissues, when ventricular contraction changes perfusion pressure to meet metabolic demands appropriately. This theory will help us to gain a new perspective in microvascular flow.

Sound radiation of cylinder in shallow water investigated by combined wave superposition method

It can be a difficult problem to precisely predict the sound field radiated from a finite elastic structure in shallow water channel because of its strong coupling with up-down boundaries and the fluid medium, whose sound field cannot be calculated directly by current methods, such as Ray theory, normal mode theory and other different methods, which are adaptable to sound fields from idealized point sources in waveguide. So far, there is no reliable prediction method to solve this kind of problem. A combined wave superposition method is proposed for such a problem, which combines the traditional wave superposition method with the transfer function in shallow water channel and the multi-physics field coupling numerical model. This method mainly consists of three sections:1) obtaining the normal velocity on the elastic structure surface in shallow water channel by the finite element method (FEM), whose FEM model includes the up-down boundaries and the completely absorbent sound boundaries in the horizontal direction; 2) getting the equivalent point source strength by traditional wave superposition method; 3) calculating the total sound field by adding up each point source field which is obtained by normal mode method. This method is verified by numerical simulation and theoretical analysis by using an imaginary and elastic spherical sound source respectively, and the results demonstrate that the method is valid and has high precision and calculating efficiency. The acoustic radiation characteristics from elastic cylindrical shells is investigated for different acoustic radiation sources, ocean environments and measurements. The cylindrical shell material is steel, whose radius and length are 3 m and 30 m respectively. The shallow water channel is an ideal waveguide with 50 m in depth, at the upper boundary, i.e., the free surface, the lower boundary is the Neumann boundary, i.e., the normal derivative of the acoustic pressure should be zero. The analysis frequency range is from 30 Hz to 200 Hz. The results show that due to a significant coupling effect of up-down direction boundaries on the sound field, the elastic structure can be equivalent to the point source only in low frequency and far field. The spatial field directivity distribution is more obvious at high frequency. The acoustic power measured by vertical line arrayis greatly influenced by ocean boundary and the depth of target.

Свободные интерфейсные и краевые колебания тонких упругих круговых цилиндрических oболочек со свободными торцами.

Свободные интерфейсные и краевые колебания тонких упругих круговых цилиндрических oболочек со свободными торцами.

Bi-orthogonality relations for fluid-filled elastic cylindrical shells: Theory, generalisations and application to construct tailored Green's matrices

The paper addresses the classical problem of time-harmonic forced vibrations of a fluid-filled cylindrical shell considered as a multi-modal waveguide carrying infinitely many waves. The forced vibration problem is solved using tailored Green's matrices formulated in terms of eigenfunction expansions. The formulation of Green's matrix is based on special (bi-)orthogonality relations between the eigenfunctions, which are derived here for the fluid-filled shell. Further, the relations are generalised to any multi-modal symmetric waveguide. Using the orthogonality relations the transcendental equation system is converted into algebraic modal equations that can be solved analytically. Upon formulation of Green's matrices the solution space is studied in terms of completeness and convergence (uniformity and rate). Special features and findings exposed only through this modal decomposition method are elaborated and the physical interpretation of the bi-orthogonality relation is discussed in relation to the total energy flow which leads to derivation of simplified equations for the energy flow components.

Investigation into the bistatic evolution of the acoustic scattering from a cylindrical shell using time-frequency analysis

The time and frequency analyses of the acoustic scattering by an elastic cylindrical shell in bistatic method show that the arrival times of the echoes and the resonance frequencies of the elastic waves propagating in and around the cylindrical shell are a function of the bistatic angle, β, between the emitter and receiver transducers. The aim of this work is to explain the observed results in time and frequency domains using time-frequency analysis and graphical interpretations. The performance of four widely used time-frequency representations, the Smoothed Pseudo Wigner-Ville (SPWV), the Spectrogram (SP), the reassignment SPWV, and the reassignment SP, are studied. The investigation into the evolution of the time-frequency plane as a function of the bistatic angle β shows that there are the waves propagating in counter-clockwise direction (labeled wave+) and the waves which propagate in clockwise direction (labeled waves−). In this paper the A, S0, and A1 circumferential waves are investigated. The graphical interpretations are used to explain the formation mechanism of these waves and the acoustic scattering in monostatic and bistatic configurations. The delay between the echoes of the waves+ and those of the waves− is expressed in the case of the circumnavigating wave (Scholte-Stoneley wave). This study shows that the observed waves at β=0° and β=180° are the result of the constructive interferences between the waves+ and the waves−. A comparative study of the physical properties (group velocity dispersion and cut-off frequency) of the waves+, the waves− and the waves observed in monostatic configuration is conducted. Furthermore, it is shown that the ability of the time-frequency representation to highlight the waves+ and the waves− is very useful, for example, for the detection and the localization of defaults, the classification purposes, etc.

Student experiment for measure of target strength

The target strength is a measurement of the reflection coefficient of a sonar target. Its value and feature is very important to detection and identification in active sonar, especially for fish by fishery acoustic students. A steel cylindrical shell is used as the target, whose outer diameter 194 mm, inner diameter 184 mm, length 500 mm, and both end closed covers with thickness 4 mm. A broadband transducer is selected to directionally project sound wave and a hydrophone receives the reflection wave from the target. The transducer and receiver are located at 5 m and 4 m from the target with the same depth 5 m in the water. When single frequency pulses are transmitted with signal interval frequency 50 Hz, the target strength is calculated by amplitude ratio of target echo and incidence wave at different frequencies. The result is compared with theoretical value for an infinite elastic cylindrical shell and finite length elastic cylindrical shell. This experiment will give a strategic guidance to understand what target strength is, how to understand it, and why we need it. [Work supported by National Natural Science Foundation of China (Grant No. 41374147).]

Vibration and stability of embedded cylindrical shell conveying fluid mixed by nanoparticles subjected to harmonic temperature distribution

Nonlinear vibration and instability of cylindrical shell conveying fluid-nanoparticles mixture flow are studied in this article. The surrounding elastic medium is modeled by Pasternak foundation. Mixture rule is used for obtaining the effective viscosity and density of the fluid-nanoparticles mixture flow. The material properties of the elastic medium and cylindrical shell are assumed temperature-dependent. Employing first order shear deformation theory (FSDT), the motion equations are derived using energy method and Hamilton\'s principal. Differential quadrature method (DQM) is used for obtaining the frequency and critical fluid velocity. The effects of different parameters such as volume percent of nanoparticles, boundary conditions, geometrical parameters of cylindrical shell, temperature change, elastic foundation and fluid velocity are shown on the frequency and critical fluid velocity of the structure. Results show that with increasing volume percent of nanoparticles in the fluid, the frequency and critical fluid velocity will be increases.

A new method for evaluating the natural frequency in radial breathing like mode vibration of double‐walled carbon nanotubes

AbstractThis paper investigates the radial breathing like mode (RBLM) vibration of double‐walled carbon nanotubes (DWCNTs) based on a new theory called doublet mechanics (DM) with a scale parameter. To this end, DWCNTs are considered as two concentric elastic thin cylindrical shells bounded together with van der Waals (vdW) forces between them. The vdW forces are modeled using Len‐Jones potential. Using DM, two coupled second order partial differential equations that govern the RBLM vibration of DWCNTs are derived. Subsequently, the equations of motion are solved to obtain the natural frequencies in the RBLM vibration. It is the first time that DM is used to derive the natural frequencies of the RBLM vibration for DWCNTs. The advantage of these analytical formulas is that unlike the other methods, they are explicitly dependent to scale parameter and chirality effect. It is shown that the chirality and scale parameter play significant role in the RBLM vibration response of DWCNTs. To validate the accuracy and efficiency of this work, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed.