Calculation of natural frequencies of fixed-free circular cylindrical shells

A theoretical analysis is discussed here in detail which is used to investigate the natural frequencies of fixed-free circular cylindrical shells. The results are compared with the experimental observations of other workers. Analytical results are, in general, in reasonably good agreement with the experimental results.The effect of the simplifying assumption of zero hoop and shear strains has been studied carefully. It is shown that for shells of large length-to-radius ratio this assumption can be incorporated resulting in a major simplification of the analysis involved without significantly affecting the accuracy of the results, whereas for shells of small length-to-radius ratio and for higher axial modes the simplified analysis must be used with care, especially for the low circumferential wave numbers because the effect there is to significantly increase the natural frequencies. In an attempt to understand the physical significance of this assumption, calculations were made to estimate the proportion of strain energy due to bending and stretching actions for various modal arrangements with and without this assumption. It was found that the effect of the assumption is to increase considerably the strain energy due to extension with little effect on the strain energy due to bending and thus increase the total strain energy, substantially in case of the shells with small length-to-radius ratio and for higher axial modes with consequent increase in the natural frequencies. In such cases the analysis without making this assumption is to be used and it is shown that this can quite easily be handled.The integrals involving characteristic beam functions have been derived in closed form.     

Exact solution for large amplitude free and forced oscillation of a thin spherical shell

The title problem is analyzed on the basis of the finite deformation theory of elasticity. The material of the shell is considered neo-Hookean. The governing equation is simplified for thin spherical shells. Exact expressions for the displacement field are derived for free oscillation and forced oscillations with prescribed pressure differences.     

On the crossing of thin shells

The crossing of two dust shells is considered as a simplified model of a collapsing thick layer of dust. We use the Israel's formalism to describe the development of two shells of dust which interact only gravitationally. The formalism is developed in both Schwarzschild and Kruskal coordinates.     

An exact general-relativity solution for the motion and intersections of self-gravitating shells in the field of a massive black hole

The motion with intersections of relativistic gravitating shells in the Schwarzschild gravitational field of a central body is considered. Formulas are derived for calculating parameters of the shells after intersection via their parameters before intersection. Such special cases as the Newtonian approximation, intersections of light shells, and intersections of a test shell with a gravitating shell are also considered. The ejection of one of the shells to infinity in the relativistic region is described. The equations of motion for the shells are analyzed numerically.     

Dynamic Modeling and Analysis of Wind Turbine Blade of Piezoelectric Plate Shell

This paper presents a theoretical analysis of vibration control technology of wind turbine blades made of piezoelectric intelligent structures. The design of the blade structure, which is made from piezoelectric material, is approximately equivalent to a flat shell structure. The differential equations of piezoelectric shallow shells for vibration control are derived based on piezoelectric laminated shell theory. On this basis, wind turbine blades are simplified as elastic piezoelectric laminated shells. We establish the electromechanical coupling system dynamic model of intelligent structures and the dynamic equation of composite piezoelectric flat shell structures by analyzing simulations of active vibration control. Simulation results show that, under wind load, blade vibration is reduced upon applying the control voltage.     

Scalar field condensation behaviors around reflecting shells in Anti-de Sitter spacetimes

We study scalar condensations around asymptotically Anti-de Sitter (AdS) regular reflecting shells. We show that the charged scalar field can condense around charged reflecting shells with both Dirichlet and Neumann boundary conditions. In particular, the radii of the asymptotically AdS hairy shells are discrete, which is similar to cases in asymptotically flat spacetimes. We also provide upper bounds for the radii of the hairy Dirichlet reflecting shells and above the bound, the scalar field cannot condense around the shell.     

多个未满l次壳层等效电子LS耦合原子态的多重谱项

给出用Matlab编程计算多个未满l次壳层的等效电子LS耦合原子态的矩阵计算方法,具体计算了4f75d电子组态LS耦合原子态的多重谱项的重数.     

MICRO-SENSING CHARACTERISTICS AND MODAL VOLTAGES OF LINEAR/NON-LINEAR TOROIDAL SHELLS

Toroidal shells belong to the shells of revolution family. Dynamic sensing signals and their distributed characteristics of spatially distributed sensors or neurons laminated on thin toroidal shell structures are investigated in this study. Spatially distributed modal voltages and signal patterns are related to the meridional and circumferential membrane/bending strains, based on the direct piezoelectricity, the Gauss theorem, the Maxwell principle and the open-circuit assumption; linear and non-linear toroidal shells are defined based on the thin shell theory and the von Karman geometric non-linearity. With the simplified mode shape functions defined by the Donnell-Mushtari-Vlasov theory, modal-dependent distributed signals and detailed signal components of spatially distributed sensors or neurons are defined and these signals are quantitatively illustrated. Signal distributions basically reveal distinct modal characteristics of toroidal shells. Parametric studies suggest that the dominating signal component results from the meridional membrane strains. Shell dimensions, materials, boundary conditions, natural modes, sensor locations/distributions/sizes, modal strain components, etc., all influence the spatially distributed modal voltages and signal generations.     

Vibrations of circular cylindrical shells with cutouts

An experimental and analytical study was carried out to examine the effect of circular cutouts on the resonant frequencies of thin cylindrical shells. The experimental results were obtained from tests performed on clamped-free aluminum cylinders and clamped ring-stiffened tri-acetyl cellulose shells with a lap-joint seam. The analytical solution was a simplified Rayleigh-Ritz type approximation. For the beam type mode, the circular cutouts had a significant influence on the frequency. For the mode with higher numbers of circumferential waves, however, the cutouts had a relatively small effect on the frequency spectra.     

Zero-point energy of N perfectly conducting concentric cylindrical shells

The zero-point (Casimir) energy of N perfectly conducting, infinitely long, concentric cylindrical shells is calculated utilizing the mode summation technique. The obtained convergent expression is studied as a function of size, curvature and number of shells. Limiting cases, such as infinitely close shells or infinite radius shells are also investigated.     

The effect of negative-energy shells on the Schwarzschild black hole

We construct Penrose diagrams for Schwarzschild spacetimes joined by massless shells of matter, in the process correcting minor flaws in the similar diagrams drawn by Dray and ’t Hooft (Commun Math Phys 99:613–625, 1985), and confirming their result that such shells generate a horizon shift. We then consider shells with negative energy density, showing that the horizon shift in this case allows for travel between the heretofore causally separated exterior regions of the Schwarzschild geometry. These drawing techniques are then used to investigate the properties of successive shells, joining multiple Schwarzschild regions. Again, the presence of negative-energy shells leads to a causal connection between the exterior regions, even in (some) cases with two successive shells of equal but opposite total energy.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Vibration of piezoelectric elements surrounded by fluid media

In this paper we analyse vibrational characteristics of piezoceramic shells surrounded by acoustic media. Main results are presented for radially polarized piezoceramic PZT5 elements of hollow cylindrical shapes. The coupling in the radial direction between the solid and the acoustic media is accounted for indirectly, via impedance boundary conditions. The model based on such impedance boundary condition approximations offers a robust simplified alternative to a full scale fluid-solid interaction modelling. By using this model, we analyse numerically the influence of the boundary conditions imposed in the axial direction for long, medium, and short (disk-like) piezoceramic elements.     

General theory of micropolar elastic thin shells

The paper formulates general hypotheses of micropolar elastic thin shells that are given asymptotic validation. Using these hypotheses and three-dimensional Cosserat (micropolar, asymmetric) theory of elasticity, general two-dimensional applied models of micropolar elastic thin shells with independent displacement and rotation fields, constrained rotation and low shear rigidity are constructed to suit dimensionless physical parameters of the shell material. The constructed micropolar shell models take into complete account transverse shear strain and related strain. Models of micropolar elastic thin plates and beams are particular cases of the constructed micropolar shell models. An axially symmetric stress-strain state problem of a hinged cylindrical micropolar shell is considered. Numerical analysis is used to demonstrate effective strength and rigidity characteristics of micropolar elastic shells.     

Duct noise radiation through a jet flow

The radiation of internal (or core) noise for aircraft turbojet or turbofan engines is studied analytically. The geometry of a typical engine is simplified for analytical considerations to a hemispherical shell with a jet flow and internal sound emanating through a circular hole on the axis. A linearized theory is used to derive a flow modified spherical wave equation. A forced separation technique is used to produce a modified Legendre equation describing the angular variation of the acoustic radiation field. Then a numerical technique is described for obtaining a general field solution by progressively imposing continuity of pressure across hemispherical shells as the solution is marched from near field to the far field. In a companion paper, numerical results are presented and compared with experimental results from a test configuration identical to that described by the theory.     

An analytical model to predict the response of fluid-filled shells to impact—a model for blunt head impacts

An approximate analytical model to predict the response of a fluid-filled shell of arbitrary thickness impacting with a solid elastic sphere is proposed and the limits of applicability of the equations developed are discussed. The model is based on combining the Hertzian contact stiffness and the effective local membrane and bending stiffness to derive implicit expressions for global impact characteristics including the duration of impact, the peak force transmitted, peak global acceleration of shell and sphere, and the resultant pressures induced in the fluid. Closed-form explicit expressions are also derived to predict whether the pressure response in the fluid will be hydrostatic or will exhibit large dynamic transients of pressure (and shear strain). It should be noted that the impact of hollow/empty shells with solid spheres, as well as the impact of shells with an elastic half-space, can be straightforwardly treated as limiting cases. The model is of obvious relevance to head impact modelling and selected parametric studies of the response of fluid-filled shells with geometric and material properties about those typical for the human head are given.     

Critical stability of almost adiabatic convection in a rapidly rotating thick spherical shell

In this work, the convection equations in the almost adiabatic approximation is studied for which the choice of physical parameters is primarily based on possible applications to the hydrodynamics of the deep interiors of the Earth and planets and moons of the terrestrial group. The initial system of partial differential equations (PDEs) was simplified to a single second-order ordinary differential equation for the pressure or vertical velocity component to investigate the linear stability of convection. The critical frequencies, modified Rayleigh numbers, and distributions of convection are obtained at various possible Prandtl numbers and in different thick fluid shells. An analytical WKB-type solution was obtained for the case when the inner radius of the shell is much smaller than the outer radius and convective sources are concentrated along the inner boundary.     

Free vibrations of laminated composite shells with uniformly distributed attached mass using higher order shell theory including stiffness effect

In this paper, free vibrations of a cross-ply composite shell with or without a uniformly distributed attached mass are analyzed using higher order shell theory. The results of free vibrations without distributed attached mass are validated by previous literatures. The stiffness effect of this distributed attached mass are also considered and compared with those well-known published results in which this effect is ignored. Various results for composite shells under a variety of conditions such as variations in the thickness of the shell, variation in the thickness of the distributed attached mass, variation in the radii of curvatures and various elasticity moduli are presented in this paper. In some cases, to verify the novel results, first-order shear deformation theory (FSDT) is also used. In this paper, parameters which influence the natural frequencies of the shells with attached mass including the stiffness of the mass are investigated. Parameters which are investigated in this paper are thickness of the shell, thickness of the distributed attached mass, elasticity moduli of the distributed attached mass and radius of curvatures of shells. Increasing the thickness or elasticity moduli of the distributed attached mass will increase the fundamental natural frequency of the shell. The effect of the stiffness of the distributed attached mass is decreased by decreasing the radii of curvatures or increasing the thickness of the shells.     

Observation of two-hole states at high excitation energy in (p,t) reactions

Bumps of two-hole states at high excitation energies were observed systematically in the triton spectra from (p, t) reactions with 52 MeV protons on nuclei in a broad range of masses. The cross sections of the bumps are almost equal for various targets with the same deep major shells. These cross sections vary discontinuously with variation of the corresponding deep major shell. About 20 to 50 % of the total expected strength is observed experimentally, if the bumps are assumed to arise from two-neutron pickup from the deep major shells. The centres of gravity of the bumps are located at excitation energies of about 7 to 9 MeV in all cases. On the other hand, the widths of the bumps change from about 3 MeV for ⁶⁶Zn to about 9 MeV for ²³⁰Th.     

Generalized asymptotic expansions for the wavenumbers in infinite flexible in vacuo orthotropic cylindrical shells

Analytical expressions are found for the wavenumbers and resonance frequencies in flexible, orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders n. The Donnell-Mushtari shell theory is used to model the dynamics of the cylindrical shell. Initially, an in vacuo cylindrical isotropic shell is considered and expressions for all the wavenumbers (bending, near-field bending, longitudinal and torsional) are found. Subsequently, defining a suitable orthotropy parameter ?, the problem of wave propagation in an orthotropic shell is posed as a perturbation on the corresponding problem for an isotropic shell. Asymptotic expressions for the wavenumbers in the in vacuo orthotropic shell are then obtained by treating ? as an expansion parameter. In both cases (isotropy and orthotropy), a frequency-scaling parameter (η) and Poisson's ratio (ν) are used to find elegant expansions in the different frequency regimes. The asymptotic expansions are compared with numerical solutions in each of the cases and the match is found to be good. The main contribution of this work lies in the extension of the existing literature by developing closed-form expressions for wavenumbers with arbitrary circumferential orders n in the case of both, isotropic and orthotropic shells. Finally, we present natural frequency expressions in finite shells (isotropic and orthotropic) for the axisymmetric mode and compare them with numerical and ANSYS results. Here also, the comparison is found to be good.