Infinite Shell Research Articles

Transient Deformation of an Anisotropic Cylindrical Shell with Structural Features

In this work, a new transient function of normal displacements is obtained in a thin elastic anisotropic cylindrical shell of constant thickness with structural features. Structural features form of two sequences in angular direction of weld spots or rivets. Structural features are mathematically described by point boundary conditions of rigid pinching or point boundary conditions of hinge support, accordingly. A load with time variable amplitude and impact boundaries influences on the outer surface of the shell along the normal. Such movement of the shell is considered in the cylindrical coordinate system as connected with the shell axis. Kirchhoff–Love hypothesis are confirmed as a shell model. The solution of this problem can be formed with the help of Green function method and lift compensation method. The required function of transient normal displacements is represented as the sum of integral operators of Green convolution for an infinite shell with functions of current transient load and compensating loads out of point boundary conditions. Compensating loads satisfying to boundary conditions can be found based on the solution of Volterra integral equations of the first kind with the difference kernel. Green function is such kernel. The system solution can be executed including preliminary discretization of compensating time loads. The convolution integrals can be taken numerically with the help of quadrature formula by the method of rectangles. The verification of the method is represented by comparing the solution of specific case with the known solution for a hinged isotropic cylindrical shell.

Coupled approximate long and short solutions versus exact Navier and Galerkin ones for cylindrical shell under radial load

Cylindrical shell under concentrated force is a fundamental problem, which provides the understanding of general patterns of deformation and allow to consider many other types of loading and geometries. The common idea of existing approaches is expansion of the looking for solutions into Fourier series with respect to nφ, where φ is a circumferential coordinate, and n is the circumferential wave number. This reduces the problem to the 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relations of 8 constants of integration with boundary conditions are still beyond the possibilities of analytical treatment. Most analytical solutions are based on Vlasov-like solution of simplified 4th order equations with unknown ranges of their accuracies. The goal of paper is threefold. First, to give the exact solutions for infinite shell, which are based on: (a) the well-known Navier approach, and (b) the Galerkin methods with application of decaying exponential functions. Second, to modernize the hypotheses of Vlasov’s theory (long-wave axial solution) and supplement it with the short solution, and commonly use both in formulation of boundary conditions, i.e., to take into account that their interaction occurs not due to shear force only but other force and displacement parameters may interact too. Third, to perform the tedious comparison of exact Galerkin and Navier solutions with approximate ones for the whole range of circumferential waves n, assess their accuracy and establish the accuracy of plate-like solution for the larger number of n.

Asymptotic Methods for Studying an Elliptical Boundary Layer in Shells of Revolution Under Normal Type Shock End Impacts

In this paper, asymmetric equations for an elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves, which occurs in shells of revolution under shock end impacts of normal type are constructed. The technique of asymptotic derivation of these equations, based on the use of the symbolic Lurie method and the introduction of special coordinates that distinguish a small frontal zone required to reduce the original problem to an equivalent problem for an infinite shell by isolating a particular solution. The considered boundary layer complements the full description of the considered type of stress-strain state (SSS) in all sections of the phase plane. It also uses a quasi-static boundary layer of the Saint-Venant type in a small vicinity of the butt end, a parabolic boundary layer according to the two-dimensional Kirchhoff–Love theory, a quasi-plane short-wave component, and a hyperbolic boundary layer in a small neighborhood of the shear wave front. In conclusion, an example of constructing an elliptical boundary layer under shock action on the butt end of a cylindrical shell is considered.

Acoustic performance prediction of a multilayered finite cylinder equipped with porous foam media

This paper presents an analytical model to embed porous materials in a finite cylindrical shell in order to obtain the sound transmission loss coefficient. Although the circumferential modes are considered only for calculating the amount of the transmitted noise through an infinitely long cylinder, the present study employs the longitudinal modes in addition to circumferential ones to analyze the vibroacoustic performance of a simply supported cylinder subjected to the porous core based on the first order shear deformation theory. To achieve this goal, the structure is immersed in a fluid and excited by an acoustic wave. In addition, the acoustic pressures and the displacements are developed in the form of double Fourier series. Since these series consist of infinite modes, it is essential to terminate this process by considering adequate modes. Hence, the convergence checking algorithm is employed in the form of some three-dimensional configurations with respect to length, frequency and radius. Afterwards, some figures are plotted to confirm the accuracy of the present formulation. In these configurations, the obtained sound transmission loss from the present study is compared with that of the infinite one. It is shown that by increasing the length of the structure, the results are approached to sound transmission loss of the infinite shells. Moreover, a new approach is proposed to show the transverse displacement of a finite poroelastic cylinder at different frequencies. Based on the outcomes, it is found that by enhancing the length of the poroelastic cylinder, the amount of the transmitted sound into the structure is reduced at the high frequency domain. However, the sound insulation property of the structure is improved at the low frequency region when the radius of the shell is decreased.

The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure

Acoustic analysis of the four-sides simply supported doubly curved composite shell interlayered with porous material used in aerospace applications is considered based on Third order Shear Deformation Theory (TSDT). The focus is specifically placed on presenting the effect of boundaries on Sound Transmission Loss (STL) of the poroelastic structure. Then, the results are compared with those of infinite shell. In order to calculate the STL, the displacement and pressure terms are derived in the form of double Fourier series for a simply supported boundary condition doubly curved shell. A new approach is made to consider the number of modes for the finite composite structure treated with porous material. In addition, the simultaneous solution of the equations of porous and composite layers along with acoustic wave equations is performed. This process leads to obtain the unknown constants in all parts of the structure including upper and bottom composite shells as well as porous core which results in presenting the STL of the structure in the logarithmic scale. Since, the solution is provided in the form of modal infinite series which should be terminated at the sufficient modes, the convergence checking is prepared in 3D configurations with respect to various frequencies and shell dimensions to present the number of the appropriate modes for this process. Moreover, the results are validated with the aid of other researches including experimental as well as analytical models available in literature. Finally, some new results including Sound Pressure Level (SPL) in both of global view and contour map, transverse displacement configurations and the effects of various shell dimensions on STL, are presented for this finite structure.

The Critical Velocity for an Infinite Cylindrical Shell under a Moving Load

The critical velocity for an infinite cylindrical shell subjected a moving load with a constant velocity is analyzed in this paper. It is found that the critical velocity problem can be translated into a distribution of the real roots of a quadruplicate equation, which can be solved by using Descartes sign method and complete discrimination system for polynomials. Our research shows that the number of the critical velocities for an infinite cylindrical shell always is even number. Furthermore the longitudinal wave velocity is not one critical velocity for the shell. Our results are different from the conclusion drawn by other authors that there are three critical velocities in an infinite shell, and the longitudinal wave velocity is the maximum critical velocity. Then further studies are needed to clarify these questions.

Critical velocity of sandwich cylindrical shell under moving internal pressure

Critical velocity of an infinite long sandwich shell under moving internal pressure is studied using the sandwich shell theory and elastodynamics theory. Propagation of axisymmetric free harmonic waves in the sandwich shell is studied using the sandwich shell theory by considering compressibility and transverse shear deformation of the core, and transverse shear deformation of face sheets. Based on the elastodynamics theory, displacement components expanded by Legendre polynomials, and position-dependent elastic constants and densities are introduced into the equations of motion. Critical velocity is the minimum phase velocity on the desperation relation curve obtained by using the two methods. Numerical examples and the finite element (FE) simulations are presented. The results show that the two critical velocities agree well with each other, and two desperation relation curves agree well with each other when the wave number k is relatively small. However, two limit phase velocities approach to the shear wave velocities of the face sheet and the core respectively when k limits to infinite. The two methods are efficient in the investigation of wave propagation in a sandwich cylindrical shell when k is relatively small. The critical velocity predicted in the FE simulations agrees with theoretical prediction.

Green’s Function for a Closed, Infinite, Circular Cylindrical Elastic Shell

An acceptable variant of the Koiter–Morley equations for an elastically isotropic circular cylindrical shell is replaced by a constant coefficient fourth-order partial differential equation for a complex-valued displacement-stress function. An approximate formal solution for the associated “free-space” Green’s function (i.e., the Green’s function for a closed, infinite shell) is derived using an inner and outer expansion. The point wise error in this solution is shown rigorously to be of relative order (h∕a)(1+h∕a∣x∣), where h is the constant thickness of the shell, a is the radius of the mid surface, and ax is distance along a generator of the mid surface.

The Effect of a Geometrical Parameter on the Deformation of a Hinged Flexible Noncircular Cylindrical Shell

The exact analytical solution of a nonlinear boundary-value problem is used to study the effect of a generalized geometric parameter, which characterizes thickness and curvature, on the subcritical and postcritical deformation of a hinged infinite noncircular cylindrical shell. The load on the shell is nonuniformly distributed over its cross section. The deflection of the shell is plotted for various values of the geometric parameter

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes ( n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.

CLOSED FORM SOLUTIONS FOR THE POINT MOBILITIES OF AXISYMMETRICALLY EXCITED CYLINDRICAL SHELLS

Closed form formulae are reported describing point mobilities for thin cylindrical shells in axisymmetric motion. Two cases are studied: (1) the infinite shell and (2) the edge-excited semi-infinite shell and, for both, forces and moment excitation are considered. The solutions of these problems are obtained analytically by resorting to perturbation methods and presented in terms of the Green functions and point mobilities. These results are further used to derive approximate expressions for the reflection coefficients of the shell-borne waves at a free end and for the Green functions of a finite free–free shell undergoing axisymmetric vibrations.

Edge and interfacial vibrations in elastic shells of revolution

Edge vibrations of a semi-infinite shell of revolution and interfacial vibrations of a longitudinally inhomogeneous infinite shell of revolution are considered. Localization of bending and extensional modes is analyzed on the basis of the Kirchhoff-Love theory. The effect of small radiation to infinity is involved. A link between localized vibrations and Rayleigh-type and Stonely-type waves is revealed. The asymptotic formulae for natural frequencies, including their imaginary parts associated with radiation, are derived.

Axisymmetric actuation of composite cylindrical thin shells with piezoelectric rings

Axisymmetric static and frequency analyses of anisotropic cylindrical thin shells with one and two perfectly bonded ring piezoactuators are performed. The shell is assumed to be linear elastic and made of laminated composite materials. The electroelastic constitutive relations for the piezoelectric materials are also assumed to be linear. It is shown that, if there exists a special relationship involving the membrane and the membrane-bending coupling stiffness matrices, the analysis is greatly simplified. In such a situation, simple closed form solutions of the equilibrium equations are obtained for the case of an infinite shell with one or two actuators. Kirchhoff's assumptions are used for the analysis and the dynamic formulation is derived from a variational principle which includes the total structural potential energy and the electrical potential energy of the piezoelectric material, involving both mechanical and electrical variables. The finite element method is then applied to obtain the stiffness and mass matrices. The computer code developed to implement the formulation allows the static and dynamic analyses of arbitrary cylindrical shells with piezoelectric actuators. Good agreement is reached between the analytical solution found and the numerical procedure implemented. Results indicate that the maximum normalized displacement and its location vary according to the actuator length. Furthermore, a frequency analysis is carried out in a broad range of frequencies to investigate the effect of mass properties on the response of a simply supported cylindrical shell.

The set valued unified model of dispersion and attenuation for wave propagation in dielectric (and anelastic) media

Since the dispersion and attenuation properties of dielectric and anelastic media, in the frequency domain, are expressed by similar formulae, as shown experimentally by Cole and Cole (1941) and Bagley and Torvik (1983, 1986) respectively, we note that the same properties may be represented in the time domain by means of an equation of the same form; this is obtained by introducing derivatives of fractional order into the system functions of the media. The Laplace Transforms (LT) of such system functions contain fractional powers of the imaginary frequency and are, therefore, multivalued functions defined in the Riemann Sheets (RS) of the function. We determine the response of the medium (dielectric o anelastic) to a generic signal summing the time domain representation due to the branches of the solutions in the RSs of the LT. It is found that, if the initial conditions are equal in all the RSs, the solution is a sum of two exponentials with complex exponents, if the initial conditions are different in some of the RSs, then a transient for each of those RSs is added to the exponentials. In all cases a monochromatic wave is split into a set of waves with the same frequency and slightly different wavelengths which interfere and disperse. As a consequence a monochromatic electromagnetic wave with frequency around 1 MHz in water has a relevant dispersion and beats generating a tunnel effect. In the atmosphere of the Earth the dispersion of a monochromatic wave with frequency around 1 GHz, like those used in tracking artificial satellites, has a negligible effect on the accuracy of the determination of the position of the satellites and the positioning of the bench marks on the Earth. We also find the split eigenfunctions of the free modes of infinite plates and shells made of dielectric and anelastic media.

Investigation of flexural waves in an infinite shell of revolution

Investigation of flexural waves in an infinite shell of revolution

The structural acoustics and active control of interior noise in a ribbed cylindrical shell

Numerical studies are carried out regarding the structural acoustics of a ribbed aluminum cylindrical shell structure intended to represent the essential structural features of a small aircraft fuselage. Calculations are made to determine both the wall normal displacements and the interior acoustic pressures for the case in which the shell wall is forced dynamically at a point. The structural responses are further decomposed into their frequency-wave-number components. Through a series of comparisons between those responses for an infinite shell, the ribbed vacuum-filled shell, and the ribbed air-filled shell, the relevant structural acoustic mechanisms are interpreted. The frequencies at which interior acoustic ‘‘resonances’’ are observed are connected to specific mechanisms, including cavity responses driven by structural modes, cavity modes forcing the structure, and mixed structure/air-cavity modes. Numerically based active control experiments are carried out using end actuation, and the relative performance of this control ‘‘system’’ is compared when operating on the various mode types.

Localized wave forms of motion of an infinite shell of revolution

The wave forms of motion of a thin infinite shell of revolution defined by initial displacements and velocities localized in the neighbourhood of a parallel are studied. Using the method developed in [1], a solution of the equations of motion is constructed as a superposition of tangential and flexural wave packets propagating in the direction of the shell axis. As an example, an elongated conical shell is considered.

Power flow between circular cylindrical shells and flat plates

An analytic model for the transmission of power between circular cylindrical shells and flat plates attached along a circumference of the shell is developed. The model includes transverse displacements associated with bending waves and in-plane displacements associated with longitudinal and shear waves. Plates internal and external to both semi-infinite and infinite shells are modeled. Examples of power transmission coefficients for different incident wave types in the plate, circumferential mode numbers and frequencies are presented and discussed.

Acoustic scattering from a circular cylindrical shell excited by a short pulse in oblique incidence: Helical waves

The theoretical and experimental spectra, obtained from an infinite pipe insonified with a plane wave in oblique incidence, show resonances which are related to three types of helical waves: the circumferential waves (Ai or Si waves), the transversal guided waves (Ti waves), and the Scholte wave (A wave). The resonances of the Scholte wave are detected in a frequency window and their frequency slowly increases when the incidence angle increases, whereas the ones of the other waves increase towards infinity when the incidence angle tends towards the transversal critical angle. At oblique incidence, the experimental results obtained with the MIIR show resonances which are related to the helical waves S0, T0, T1, and A. This method uses a long pulse with many sinusoid periods, a steady state takes place in a part of the infinite shell. In this presentation, the excitation is a short pulse and the scattered echoes are detected when the helical wave emission is in front of the receiver. Between each echo, the helical wave has covered one step of the helix (L=2πa/cos γ, a is the outer radius of cylinder, γ is the angle of the helix). This method allows us to calculate the group velocity of the helical waves.

Sound scattering of an arbitrary wave incident on a simply-supported cylindrical shell.

Acoustic scattering of an arbitrary wave incident on a finite submerged cylindrical shell is studied theoretically and experimentally, where the term arbitrary wave is referred to non-plane, non-spherical wave-front emanating from an actual transducer driven by a single frequency. An analytical solution for the spherical wave incidence is extended by introducing the concepts of “insonified window” and “shape parameter.” A simplysupported cylindrical shell for the experiment may be modeled as an infinite shell whose insonified length is restricted by the insonified window. The shape parameter defines the difference in the shape of wave-front between the spherical wave and the wave from an actual transducer. The measurement of scattering on a simply-supported, waterfilled shell is carried out using Scattering Generalized Near-field Acoustical Holography (SGENAH). The validity of our approximate theory is confirmed by good agreement between theoretical and experimental results.