Motion Of Shell Research Articles

Numerical Investigation of Shell Formation in Thin Slab Casting of Funnel-Type Mold

The key issue for modeling thin slab casting (TSC) process is to consider the evolution of the solid shell including fully solidified strand and partially solidified dendritic mushy zone, which strongly interacts with the turbulent flow and in the meantime is subject to continuous deformation due to the funnel-type mold. Here an enthalpy-based mixture solidification model that considers turbulent flow [Prescott and Incropera, ASME HTD, 1994, vol. 280, pp. 59–69] is employed and further enhanced by including the motion of the solidifying and deforming solid shell. The motion of the solid phase is calculated with an incompressible rigid viscoplastic model on the basis of an assumed moving boundary velocity condition. In the first part, a 2D benchmark is simulated to mimic the solidification and motion of the solid shell. The importance of numerical treatment of the advection of latent heat in the deforming solid shell (mushy zone) is specially addressed, and some interesting phenomena of interaction between the turbulent flow and the growing mushy zone are presented. In the second part, an example of 3D TSC is presented to demonstrate the model suitability. Finally, techniques for the improvement of calculation accuracy and computation efficiency as well as experimental evaluations are also discussed.

Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer

This paper addresses the nonlinear vibration problem of simply supported functionally graded (FG) cylindrical shells with embedded piezoelectric layers. The governing differential equations of motion of the FG cylindrical shell are derived using the Lagrange equations under the assumption of the Donnell׳s nonlinear shallow-shell theory. A semi analytical approach, wherein the displacement fields are expanded by means of a double mixed series based on linear mode shape functions for the longitudinal, circumferential and radial variables, is proposed to characterize the nonlinear response of the cylindrical shell. The large-amplitude response and amplitude frequency curves of the cylindrical shell are obtained by using the proposed approach. Finally, the effects of excitation force and applied voltage on the vibration behavior of the cylindrical shell are investigated.

Cylindrical thin-shell model based on modified strain gradient theory

In this paper, the cylindrical thin-shell model is developed based on modified strain gradient theory. For this purpose, the study develops the thin shell theory, having considered size effects through modified strain gradient theory. Besides, partial equations of shell motion with classical and non-classical corresponding boundary conditions are derived from Hamilton principle. Finally, by way of example, the free vibration of the single-walled carbon nanotube (SWCNT) is investigated. The study models the SWCNT as a simply-supported shell. Besides, the Navier procedure is used to solve the vibration problem. The results of the new model are compared with those of the couple stress model and the classical theory, leading to the conclusion that the mentioned models are special cases of the modified strain gradient theory. The findings also indicate that the rigidity of the nanoshell in the modified strain gradient theory is greater than that in couple stress model and the classical theory, which leads to the increase in natural frequencies. Furthermore, the effect of the material length scale parameter on the vibration of the nanoshell for different lengths is taken into account.

Investigation of free oscillations of the spherical shell containing a liquid by the inverse method

The problems of free oscillations of shells, contacting with the continuous medium are considered in the known works of various authors. As a rule, the problems are reduced to the transcendental equations or the systems the solution of which by analytical methods is not possible. The results of the investigation are presented in the form of tables or graphs, obtained by numerical methods. The problem of axisymmetric free oscillations of elastic thin-walled spherical shell, containing a compressible liquid is considered in the paper. Herewith, the equations of motion are constructed in radial motions and with the use of special potential. The problem is reduced to the investigation of the homogeneous system of two equations with respect to the radial motion and the mentioned potential. The condition of non-triviality of the system solution leads to the transcendental equation. In the known works, the solution of the specified transcendental equations is found by numerical methods. The analytical solution, binding the frequency of the system shell - liquid with the frequency of the shell without the liquid is constructed by the inverse method. This solution allows to investigate the phenomenon of the analytical method and to build the frequency spectra. The potential motion both of shell and liquid is considered. The equation of the shell motion in special potentials is used. The liquid motion equation is represented by the wave equation. The liquid motion is proposed as non-separable.

Numerical modeling of elliptical cylindrical shells, containing flowing fluid

AbstractThe paper presents a finite‐element algorithm for investigation of the dynamic behavior of elastic shells with arbitrary cross sections containing a flowing fluid. A curvilinear surface of the structure is approximated by a set of plane elements, which are subjected to both membrane and bending forces. Description of the shell motion is based on the variable principle of virtual displacements, which includes the linearized Bernoulli equation for calculation of hydrodynamic pressure. A compressible non‐viscous fluid is considered in the framework of the potential theory, the equations of which are transformed with the use of the Bubnov‐Galerkin method. The performance of the algorithm is demonstrated by the example of two shells with a circular and elliptical cross section. Numerical simulation have been carried out to analyze the influence of the ratio of the ellipse semi‐axes, the linear dimensions of the structure and the level of shell filling on the hydrodynamic stability of the connected “shell‐fluid” system. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear Dynamics of Cantilever FGM Cylindrical Shell under 1:2 Internal Resonance Relations

An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal and static in-plane pre-applied excitation is presented in a thermal environment. Material properties are assumed to be temperature-dependent. Based on Reddy's first-order shell theory, the nonlinear governing equations of motion for the FGM cylindrical shell are derived using Hamilton's principle. Galerkin's method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. The effects of external excitations on the internal resonance relationship and nonlinear dynamic response of FGM cylindrical shell are studied. The resonant case considered here is 1:2 internal resonance. A numerical method is used to find the nonlinear dynamic responses of the cantilever FGM cylindrical shell with different volume fraction index. It is found that the vibration amplitudes are not very sensitive to the value of the volume fraction index in the cases of 1:2 internal resonance. With the increasing of the volume fraction index, the value of and increased remarkably. There is a boundary value of the forcing amplitude, which divides the response into two parts for the case of different volume fraction index. One is the multiple periodic motion of the FGM cantilever cylindrical shell and the other is the quasi-period response. A larger value of volume fraction index leads to a larger region of quasi-period response.

Optimization of the parameters of plasma liners with zero-dimensional models

The efficiency of conversion of the energy stored in the capacitor bank of a high-current pulse generator into the kinetic energy of an imploding plasma liner is analyzed. The analysis is performed by using a model consisting of LC circuit equations and equations of motion of a cylindrical shell. It is shown that efficient energy conversion can be attained only with a low-inductance generator. The mode of an "ideal" load is considered where the load current at the final stage of implosion is close to zero. The advantages of this mode are, first, high efficiency of energy conversion (80%) and, second, improved stability of the shell implosion. In addition, for inertial confinement fusion realized by the scheme of a Z pinch dynamic hohlraum, not one but several fusion targets can be placed in the cavity on the pinch axis due to the large length of the liner.

Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations

In the present study nonlinear static and dynamic responses of shallow spherical shells resting on Winkler–Pasternak elastic foundations are carried out. The formulation of the shells is based on the Donnell theory. The nonlinear governing equations of motion of shallow shells are discretized in space and time domains using the discrete singular convolution and the differential quadrature methods, respectively. The validity of the present method is demonstrated by comparing the present results with those available in the open literature. The effects of the Winkler and Pasternak foundation parameters on nonlinear static and dynamic response of shells are investigated. Some results are also presented for circular plate as special case. Damping effect on nonlinear dynamic response of shells is studied. It is important to state that the increase in damping parameter causes decrease in the dynamic response of the shells. It is shown that the shear parameter of the foundation has a significant influence on the dynamic and static response of the shells. Also, the response of the shell is decreased with the increasing value of the shear parameter of the foundation. Parametric studies considering different geometric variables have also been investigated.

Burrowing behaviour of robotic bivalves with synthetic morphologies

Several bivalve species burrow into sandy sediments to reach their living position. There are many hypotheses concerning the functional morphology of the bivalve shell for burrowing. Observational studies are limited and often qualitative and should be complemented by a synthetic approach mimicking the burrowing process using a robotic emulation. In this paper we present a simple mechatronic set-up to mimic the burrowing behaviour of bivalves. As environment we used water and quartz sand contained in a glass tank. Bivalve shells were mathematically modelled on the computer and then materialized using a 3D printer. The burrowing motion of the shells was induced by two external linear motors. Preliminary experiments did not expose any artefacts introduced to the burrowing process by the set-up. We tested effects of shell size, shape and surface sculpturing on the burrowing performance. Neither the typical bivalve shape nor surface sculpture did have a clear positive effect on burrowing depth in the performed experiments. We argue that the presented method is a valid and promising approach to investigate the functional morphology of bivalve shells and should be improved and extended in future studies. In contrast to the observation of living bivalves, our approach offers complete control over the parameters defining shell morphology and motion pattern. The technical set-up allows the systematic variation of all parameters to quantify their effects. The major drawback of the built set-up was that the reliability and significance of the results was limited by the lack of an optimal technique to standardize the sediment state before experiments.

Dynamic Model of Elastically Mounted Machinery with Cylindrical Shell

Mounting machinery by isolators can reduce vibration transmitted to the base and attenuated environmental noise. In this paper the machine having cylindrical shell such as electric motors is modeled by thin-wall cylindrical shell motion equation. The reaction force exerted by isolator is considered as point force and integrated in the shell equation. The typical vibration excitation of machinery is represented by point and line excitations. The forces transmitted to the base through isolators are then calculated under different excitations. Conclusions with respect to machinery and isolation system design are presented based on numerical results.

Effects of Joint Discontinuity on the Vibration Power Flow Propagation in a Submerged Cylindrical Shell

The propagation characteristics of the vibration power flow in a submerged cylindrical shell with joint discontinuity are investigated by the wave propagation approach. The motion of the cylindrical shell and the pressure field in fluid are described by the Flügge shell theory and the Helmholtz equation, respectively. And the dynamic equations of the system are obtained by the coupling between the shell and the fluid. Then, an analysis of the vibration power flow transmission and reflection at the joint discontinuity is presented and the power flow transmission ratio Tr through the joint discontinuity is studied. Results show that the joint discontinuity can reduce the mean value of the Tr and thus, reduce the energy level of the transmitted vibration, as it has the effect of partially reflecting some of the incident wave with relations to its physical and geometric parameters. The influences of the fluid and the material damping of the joint discontinuity are also studied.

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.

Efficiency of pulse high-current generator energy transfer into plasma liner energy

The efficiency of capacitor-bank energy transfer from a high-current pulse generator into kinetic energy of a plasma liner has been analyzed. The analysis was performed using a model including the circuit equations and equations of the cylindrical shell motion. High efficiency of the energy transfer into kinetic energy of the liner is shown to be achieved only by a low-inductance generator. We considered an “ideal” liner load in which the load current is close to zero in the final of the shell compression. This load provides a high (up to 80%) efficiency of energy transfer and higher stability when compressing the liner.

Investigation on Vibro-Acoustic Characteristics from Submerged Cylindrical Shell with Double Damping Layers

The vibration and sound radiation from submerged cylindrical shell with double damping layers are presented. The cylindrical shell motion was described with classical thin shell theory. The double damping layers motion was described with the Navier viscoelasticity theory. For different Youngs modulus parameters of double damping layers, the sound radiated power and the radial quadratic velocity of cylindrical shell models were calculated and analyzed. The results show that the sound radiated power and radial quadratic velocity are reduced to varying degrees due to double damping layers in a large frequency domain except low frequency. The double damping layer with soft inner layer and hard outer layer can make the sound radiated peaks move to high frequency, can help to reduce the radial quadratic velocity on outer surface of damping layer, and can help to reduce the vibration of model at antiresonance frequency.

Nonlinear transient response of fibre metal laminated shallow spherical shells with interfacial damage under unsteady temperature fields

This paper is concerned with the transient response of fibre metal laminated (FML) shallow spherical shells with interfacial damage subjected to the unsteady temperature field. An exact displacement field which satisfies the boundary conditions on the outer and inner surfaces and stress continuity conditions at interfaces is presented. The nonlinear governing equations of motion for FML shallow spherical shells including the transverse shear deformation are established using the Hamilton’s principle. The transient temperature is determined from the heat conduction equation by using the finite difference method. The collocation point method and Newmark method are adapted to solve the governing equations of motion numerically. The present model provides an effective method for nonlinear dynamic analysis of composites laminated structures with interfacial damage subjected to transient temperature fields. In the numerical examples, the transient temperature, thermal deformation and stresses of FML shallow spherical shells are presented. The transient temperature will increase from initial value with time to a steady state. The displacements and stresses of the shell increase with time and remain unchanged when the temperature is in a steady state. And the effects of interfacial damage on the mechanical behaviours of FML shallow spherical shells are also discussed.

Nonlinear vibration response and bifurcation of circular cylindrical shells under traveling concentrated harmonic excitation

The nonlinear vibration of a cantilever cylindrical shell under a concentrated harmonic excitation moving in a concentric circular path is proposed. Nonlinearities due to large-amplitude shell motion are considered, with account taken of the effect of viscous structure damping. The system is discretized by Galerkin's method. The method of averaging is developed to study the nonlinear traveling wave responses of the multi-degrees-of-freedom system. The bifurcation phenomenon of the model is investigated by means of the averaged system in detail. The results reveal the change process and nonlinear dynamic characteristics of the periodic solutions of averaged equations.

Meshless analysis of shear deformable shells: the linear model

This work develops a kinematically linear shell model departing from a consistent nonlinear theory. The initial geometry is mapped from a flat reference configuration by a stress-free finite deformation, after which, the actual shell motion takes place. The model maintains the features of a complete stress-resultant theory with Reissner-Mindlin kinematics based on an inextensible director. A hybrid displacement variational formulation is presented, where the domain displacements and kinematic boundary reactions are independently approximated. The resort to a flat reference configuration allows the discretization using 2-D Multiple Fixed Least-Squares (MFLS) on the domain. The consistent definition of stress resultants and consequent plane stress assumption led to a neat formulation for the analysis of shells. The consistent linear approximation, combined with MFLS, made possible efficient computations with a desired continuity degree, leading to smooth results for the displacement, strain and stress fields, as shown by several numerical examples.

A sampling surfaces method and its application to three-dimensional exact solutions for piezoelectric laminated shells

The application of the sampling surfaces (SaS) method to piezoelectric laminated composite plates is presented in a companion paper (Kulikov, G.M., Plotnikova, S.V., Three-dimensional exact analysis of piezoelectric laminated plates via sampling surfaces method. International Journal of Solids and Structures 50, http://dx.doi.org/10.1016/j.ijsolstr.2013.02.015). In this paper, we extend the SaS method to shells to solve the static problems of three-dimensional (3D) electroelasticity for cylindrical and spherical piezoelectric laminated shells. For this purpose, we introduce inside the nth layer In not equally spaced SaS parallel to the middle surface of the shell and choose displacements of these surfaces as basic kinematic variables. Such choice of displacements permits, first, the presentation of governing equations of the proposed piezoelectric shell formulation in a very compact form and, second, gives an opportunity to utilize the strain–displacement equations, which precisely represent all rigid-body shell motions in any convected curvilinear coordinate system. It is shown that the developed piezoelectric shell formulation can be applied efficiently to finding of 3D exact solutions for piezoelectric cross-ply and angle-ply shells with a specified accuracy using a sufficient number of SaS, which are located at Chebyshev polynomial nodes and layer interfaces as well.

Charged shell supported by a phantom energy

This paper discusses the evolution of a thin spherically symmetric self gravitating phantom shell around the charged shell. The general equations describing the motion of shell with a general form of equation of state are derived. The different types of space-time R± and T± regions and shell motion are classified depending on the parameters of the problem.

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.