Influence of thickness shear deformation on free vibrations of rectangular plates,cylindrical panels and cylinders of antisymmetric angle-ply construction

This paper is concerned with the influence of thickness shear deformation and rotatory inertia on the free vibrations of antisymmetric angle-ply laminated circular cylindrical panels. Two kinds of thickness shear deformable shell theories are considered. In the first one, uniformly distributed thickness shear strains through the shell thickness and, therefore, thickness shear correction factors are used. In the second theory a parabolic variation of thickness shear strains and stresses with zero values at the inner and outer shell surfaces is assumed. The analysis is mainly based on Love's approximations but, for purposes of comparison, Donnell's shallow shell approximations are also considered. For a simply supported panel, the equations of motion of the aforementioned theories, as well as of the corresponding classical theories, are solved by using Galerkin's method. For a family of graphite-epoxy angle-ply laminated plates and circular cylindrical panels, numerical results are obtained, compared and discussed and some interesting conclusions are made regarding the shell theories considered as well as the mathematical method employed.     

Vibration of cylindrical shells of bimodulus composite materials

A theory is formulated for the small amplitude free vibration of thick, circular cylindrical shells laminated of bimodulus composite materials, which have different elastic properties depending upon whether the fiber-direction strain is tensile or compressive. The theory used is the dynamic, shear deformable (moderately thick shell) analog of the Sanders best first approximation thin shell theory. By means of tracers, the analysis can be reduced to that of various simpler shell theories, namely Love's first approximation, and Donnell's shallow shell theory. As an example of the application of the theory, a closed form solution is presented for a freely supported panel or complete shell. To validate the analysis, numerical results are compared with existing results for various special cases. Also, the effects of the various shell theories, thickness shear flexibility, and bimodulus action are investigated.     

An analytical quasi-exact method for calculating eigenvibrations of thin circular cylindrical shells

A method for calculating the eigenfrequencies and corresponding deformation modes of a thin circular cylindrical shell is presented, based on analytical solutions of Flügge's shell theory equations. The partial differential equations are transformed into algebraic equations which can be solved with high accuracy. Consequently, the results can be considered as quasi-exact. Results of calculations are presented for a shell stiffened at both circular edges. Such boundary conditions are typical of the core barrel of a pressurized water reactor, for instance. Most of the calculated deformation modes show strong gradients of the displacements close to the boundaries.     

Natural frequencies of shallow cylindrically curved panels

The natural frequencies of cylindrically curved panels are available in closed form for only two boundary condition sets. This paper demonstrates how Sewall's shallow shell formation can be recast in a relatively simple form to allow direct computation of the natural frequencies and mode shapes of cylindrical panels with a wide range of boundary conditions.     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

STRUCTURAL RESPONSE OF LAMINATED COMPOSITE SHELLS SUBJECTED TO BLAST LOADING: COMPARISON OF EXPERIMENTAL AND THEORETICAL METHODS

The results from a theoretical and experimental investigation of the dynamic response of cylindrically curved laminated composite shells subjected to normal blast loading are presented. The dynamic equations of motion for cylindrical laminated shells are derived using the assumptions of Love's theory of thin elastic shells. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. The time-dependent equations of the cylindrically curved laminated shell are then solved by the Runge-Kutta-Verner method. Finite element modelling and analysis for the blast-loaded cylindrical shell are also presented. Experimental results for cylindrically curved laminated composite shells with clamped edges and subjected to blast loading are presented. The blast pressure and strain measurements are performed on the shell panels. The strain response frequencies of the clamped cylindrical shells subjected to blast load are obtained using the fast Fourier transformation technique. In addition, the effects of material properties on the dynamic behaviour are examined. The strain-time history curves show agreement between the experimental and analysis results in the longitudinal direction of the cylindrical panels. However, there is a discrepancy between the experimental and analysis results in the circumferential direction of the cylindrical panels. A good prediction is obtained for the response frequency of the cylindrical shell panels.     

Determination of rs coordinates for atoms on a principal plane or principal axis of inertia of a molecule

For the location of an atom on a principal plane or principal axis of inertia of a molecule Chutjian's version of Kraitchman's equations is recommended because it takes proper account of the symmetry of the problem without explicitly requiring the out-of-plane or out-of-axis coordinates to vanish, i.e., without using the pseudoinertial defect relations which are only approximately satisfied by ground-state moments of inertia. It is shown that a coordinate determined in this way is unambiguous and is the mean of the scattering values produced when the vanishing of the pseudoinertial defect differences on substitution is used to obtain “specialized forms” of Kraitchman's equations.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Free vibration analysis of circular cylindrical shells

A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.     

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.     

Shells and plates loaded by dynamic moments with special attention to rotating point moments

The response of thin shells to line or point moment excitation is formulated by way of distributed moment fields. Twisting moments in the tangent plane are part of this formulation. The approach is illustrated by using Love's thin shell theory, but is valid for any other thin shell theory as well. Dirac delta functions are used to describe line and point moments. As a first example, the response of a plate to a rotating moment is evaluated and shown to be identical to the solution obtained by Bolleter and Soedel [1] by a Green function approach. The three-directional response of a circular cylindrical shell to a rotating moment is given as the second example. It is a technically significant case that has not been treated in the literature before.     

Complex symmetries of electrodynamics

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincaré group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincaré transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

An integral equation analysis of the harmonic response of three-layer beams

The integral equations of harmonic motion have been derived and solved for three-layer sandwich beams with a constrained linear viscoelastic core. The method of solution required first the construction of the Green's vector for a beam in analytical form. Following this, the integral equations were derived and readily approximated by matrix equations which were finally solved numerically. In addition to this analysis, the corresponding eigenvalue problem has been solved so that the modal frequencies and the beam loss factor could be calculated directly. The integral equation analysis offers a fast and efficient alternative to the traditional methods based on the solution of the differential equations of motion. The method has been verified by comparison with experimental results for three-layer cantilevers and simply supported beams.     

On the structure of the three-particle operator in generalized kinetic equations for inhomogeneous gases

It is shown that the three-particle kinetic operator for inhomogeneous gases obtained using Prigogine's method and the matrix representation of the Liouville equation introduced by Balescu is equivalent to the corresponding expression derived by Choh and Uhlenbeck using Bogolubov's method. Both theories take into account the space and time delocalization associated with finite collision time, and the resulting corrections to the asymptotic collision operator are equivalent.     

Circumferential waves for a cylindrical shell in smooth contact with a continuum

Dispersion relations are determined for circumferential waves propagating in a layered, circular cylinder by using shell equations to approximate the behavior of the outer layer. These equations include the effects of transverse shear deformation and rotatory inertia. The cylinder consists of an elastic core in smooth contact with a hollow, circular cylinder of distinctly different elastic properties. Two distinct modes exist as the shell thickness reduces to zero. One mode is recognized to be surface waves on the convex cylindrical surface of the core; the second mode is associated with long longitudinal waves in the shell. The approximate dispersion curves for these modes are compared with curves obtained by employing elasticity equations for the layer. As the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better when the layer is relatively stiff than when the layer is relatively soft. The shell equations simplify the calculations necessary to produce the dispersion curves.     

Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer

A method is presented to predict the root mean square displacement response of an open curved thin shell structure subjected to a turbulent boundary-layer-induced random pressure field. The basic formulation of the dynamic problem is an efficient approach combining classic thin shell theory and the finite element method, in which the finite elements are flat rectangular shell elements with five degrees of freedom per node. The displacement functions are derived from Sanders’ thin shell theory. A numerical approach is proposed to obtain the total root mean square displacements of an open curved thin structure in terms of the cross spectral density of random pressure fields. The cross spectral density of pressure fluctuations in the turbulent pressure field is described using the Corcos formulation. Exact integrations over surface and frequency lead to an expression for the total root mean square displacement response in terms of the characteristics of the structure and flow. An in-house program based on the presented method was developed. The total root mean square displacements of a curved thin blade subjected to turbulent boundary layers were calculated and illustrated as a function of free stream velocity and damping ratio. A numerical implementation for the vibration of a cylinder excited by fully developed turbulent boundary layer flow was presented. The results compared favorably with those obtained using software developed by Lakis and Païdoussis (J. Sound Vib. 25 (1972) 1–27) using cylindrical elements and a hybrid finite element method.     

Poisson finite difference boundary equations for the magnetic potentials

Magnetostatic solutions may be obtained by using Laplace's equation in finite difference form. The boundary equations are of primary importance and even these have been put forward as special forms of Laplace's equation. They can be derived in what appears to be a simpler and more flexible manner by assuming that Poisson's equation applies at the interface for both scalar and vector potentials.     

Method of Green’s function of nonlinear vibration of corrugated shallow shells

Based on the dynamic equations of nonlinear large deflection of axisymmetric shallow shells of revolution, the nonlinear free vibration and forced vibration of a corrugated shallow shell under concentrated load acting at the center have been investigated. The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green’s function. To solve the integral-differential equations, the expansion method was used to obtain Green’s function. Then the integral-differential equations were reduced to the form with a degenerate core by expanding Green’s function as a series of characteristic function. Therefore, the integral-differential equations became nonlinear ordinary differential equations with regard to time. The amplitude-frequency relation, with respect to the natural frequency of the lowest order and the amplitude-frequency response under harmonic force, were obtained by considering single mode vibration. As a numerical example, nonlinear free and forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied. The obtained solutions are available for reference to the design of corrugated shells.