Nonaxisymmetric Problem Research Articles

Application of fourier-series methods and integral equations for solving nonstationary nonaxisymmetric heat conduction problems for bodies of revolution

We consider the problem of determining nonstationary nonaxisymmetric temperature fields in bodies of revolution appearing on heating by internal heat sources through and due to convective heat exchange with an external medium. The solution of the problem is represented in the form of a Fourier series in an angular coordinate with coefficients being determined by a method of boundary elements. We consider the general case and particular cases of the nonstationary nonaxisymmetric heat conduction problem and determine the asymptotic temperature distributions with a linear variation in time of the heating medium temperature and with heating by moving heat sources.

Elastic wave scattering by an inclusion in a multilayered medium submerged in fluid. II. Three-dimensional model

Response of plane three-dimensional multilayered medium enclosing an elastic inclusion of arbitrary shape and submerged in fluid is investigated. The system is subjected to an oblique plane incident wave from within the fluid. The steady-state free-field response is modeled by the Thomson-Haskell method using delta-matrix modification and the scattered wave field by a boundary method. The transient response is evaluated using the Fourier synthesis. Numerical results are presented for a single layer plate with an ellipsoidal inclusion. The steady-state results show that the response along the fluid-solid interface is sensitive to the aspect ratio and location of the inclusion, impedance contrast, and the angle of incidence. In addition, the results of the three-dimensional analysis for the nonaxisymmetric problem indicate significant variation of the radial and azimuthal displacement components with the change in the azimuthal angle. Comparison of the two-dimensional and three-dimensional models indicates that for the harder inclusion, both the models show similar qualitative trends in the transient and steady-state responses. However, for a softer inclusion, the steady-state responses of the two models are found to differ significantly from each other. The transient studies clearly indicate the horizontal wave traveling at the solid-fluid interface with the acoustic wave velocity of the fluid. The effect of the scatterer on the vertical displacement is found to be significant only in the vicinity of the inclusion.

Extension of the surface variational principle to nonaxisymmetric response of submerged thin shells of revolution

Previously, the surface variational principle (SVP) was implemented for rigid and elastic submerged structures under axisymmetric excitations. More recent developments extended SVP to address nonaxisymmetric problems of acoustic radiation and scattering situations for rigid bodies of revolution. In this presentation, SVP is applied jointly with Hamilton’s principle governing nonaxisymmetric responses of elastic thin shells of revolution. Complex Fourier expansions of the azimuthal dependence are combined with Ritz expansions of the spatial dependence to represent the displacement field and surface pressure. This representation is equivalent to a wave-number decomposition of the surface response into a series of helical-type waves. By enforcing continuity of normal velocity on the fluid–structure interface, a set of coupled algebraic equations is formed for each azimuthal harmonic. Problems addressed here are radiation due to arbitrary loading, and scattering associated with arbitrary incidence of a plane wave on a stationary elastic thin shell. The rigid body modes are included in the scattering case. Numerical results are assessed for the convergence qualities of an SVP simulation, particularly regarding series length requirement as the frequency is increased. [Work support by ONR.]

Flow in a spherical cavity due to a stokeslet

The earth is considered as a rigid spherical cavity of radius a filled with a highly viscous incompressible fluid of viscosity η. The non-axisymmetric problem of flow due to a stokeslet of strength F/8πη located at (0, 0, c), c < a with its axis along OX, O being the centre of the sphere, is discussed for small Reynolds numbers. The expressions for velocity and pressure are obtained in terms of A(r, θ, ϕ) and B(r, θ, ϕ), biharmonic and harmonic functions respectively, using a sphere theorem for non-axisymmetric flow inside a sphere. The drag on the sphere exerted by the fluid is found to be independent of the location of the stokeslet.

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.

Experimental and theoretical study of toroidal vesicles

We report the observation of toroidal and higher genus vesicles of diacetylenic phospholipids, a class of polymerizable amphiphiles. When unpolymerized, the vesicles exhibit different toroidal shapes in quantitative agreement with recent theoretical predictions. When partially polymerized, only a specific family of shapes has been observed: the Clifford torus or the branch of nonaxisymmetric shapes obtained by its conformal transformations. Assuming that partially polymerized vesicles are permeable on short time scale, we give a physical explanation of our findings. We also report the results of a variational calculation which approximates the nonaxisymmetric shape problem for finite spontaneous curvature.

The Point-Load Solution Using Linearized von Kármán Plate Theory for a Spinning Flexible Disk Near a Baseplate

In magnetic and/or optical recording on flexible media, an elastic disk rotates at constant angular velocity in close proximity to a stationary baseplate. Such a configuration is used to stabilize the transverse motion of the flexible disk whose natural frequencies and critical speeds would otherwise be too low for stability of the flexible-disk-to-head interface. The influences of coupling between the in-plane displacements and transverse deflections (von Karman plate theory) and of the air flow between the disk and the baseplate are investigated. Steady-state solutions, for this nonaxisymmetric problem, are obtained by linearizing the partial differential equations about the axisymmetric configuration due to the baseplate alone. These equations are solved using an exponential Fourier series expansion in the circumferential direction and a finite difference approximation radially. The results are the transverse and in-plane deflections of the disk and the air film pressure. Comparisons are made with othe...

The non-axisymmetric problem of the stress concentration in an unbounded elastic medium near a spherical slit

Using the approach described in [1], the non-axisymmetric problem of the stress concentration in an unbounded elastic medium near a spherical slit is reduced to three, one-dimensional, separately solvable, integral equations. Their exact solution is obtained in a class of functions with non-integrable singularities and is used, as in [1], to derive simple formulae for the stress intensity factors. The axisymmetric modification of this problem has been investigated in [2–5].

Mixed boundary value problem of potential theory in toroidal coordinates

A new approach is presented for solving the title problems. The method presents further extension of previously obtained results to the case of toroidal coordinates. The method allows us to solve non-axisymmetric problems exactly and in closed form, with no integral transform or special function expansions involved. Some integrals of fundamental value, involving distances between several points, are established.

Use of translational addition theorems for spherical wave functions in nonaxisymmetric acoustic field problems

The translational addition theorems for spherical wave functions are used to develop solutions for nonaxisymmetric acoustic field problems involving nonconcentric spherical boundaries. The forms of the addition theorems are discussed along with their use for translating modal expansions of spherical wave functions centered on one origin to another. In a typical problem involving a number of arbitrarily positioned spherical sources within a spherical‐shaped region, an infinite set of equations for the modal expansion coefficients is developed. This set must be truncated and then solved numerically. Assuming lossless media, comparison of the near‐field and far‐field‐radiated acoustic power provides a test of the accuracy of the numerical results. Other quantities of interest such as the radiation impedance of each source, including the mutual effects of all the other sources, and directivity patterns can be readily computed once a sufficient number of modal coefficients have been computed. In a companion paper [S. A. Lease and W. Thompson, Jr., J. Acoust. Soc. Am. 90, xxx–xxx (1991)] numerical results are presented for the specific case of two identical monopole sources within a fluid sphere that is embedded in another infinite fluid medium.

A non-axisymmetric contact problem in the case of a normal load applied outside the area of contact

A non-axisymmetric contact problem in the case of a normal load applied outside the area of contact

Internal instability of laminated composites with a metal matrix

The constructed characteristic determinant for a three-dimensional nonaxisymmetric problem in the theory of stability of laminated composites with a metal matrix coincides with the characteristic determinant for the axisymmetric problem with the corresponding substitution of the wave parameters. The solution of the characteristic equation for a real material shows that loss of stability according to the different forms and for different values of deformation and the oscillation parameter as a function of the concentration of filler is possible.

Continuum approximation in three-dimensional nonaxisymmetric problems of the stability theory of laminar compressible composite materials

In the present work it has been rigorously proven that, for three-dimensional nonaxisymmetric stability problems of laminar compressible composite materials, forms of stability loss with a period along the axis ox3 larger than the period of the structure (the second and fourth forms) in the continuum approximation do not give results corresponding to internal instability; forms of stability loss with the period along the axis ox3 equal to the period of the structure (the first and third forms) in the continuum approximation give results corresponding to internal instability; the continuum theory of internal instability [2, 3] follows in the long-wave approximation, from the results corresponding to the first form of stability loss within the framework of the model of a piecewise-homogeneous medium (accurate formulation), and hence the continuum theory [2] is asymptotically accurate.

Apparent horizons for boosted or spinning black holes.

As part of an ongoing study of initial data for black-hole collisions, we examine the apparent horizons in initial-data sets for a single black hole with either translational or rotational velocity constructed using a systematic and easily generalized formalism. The apparent-horizon equation is formulated as a boundary-value problem that, with small changes, will be applicable to nonaxisymmetric problems. We find all apparent horizons in our numerically generated sets of initial data. A previously known exact result for spinning holes is reproduced numerically. We find new results, both exact and numerical, for the apparent horizons of black holes with linear momentum. In some of these cases an interesting structure emerges in which the apparent horizons and minimal surface intersect one another.

On a class of exact solutions of a non-axisymmetric contact problem for an inhomogeneous elastic half-space

A non-axisymmetric mixed boundary-value problem is considered concerning the pressure (in the absence of friction and adhesion forces) of a stiff circular-planform stamp with a base of aribitrary shape on an inhomogeneous elastic half-space. The shear modulus of the half-space material is constant while Poisson's ratio is an arbitrary piecewise-continuous function of the depth. By using the theory of dual integral equations associated with the generalized Hankel integral operator, the problem is reduced to a sequence of one-dimensional Fredholm integral equations of the second kind. It is shown that the integral equations obtained allow exact solutions to be constructed for periodic law of variation of the half-space material elastic properties with depth. The solution of a non-axisymmetric problem regarding the eccentric impression of a stamp with a flat base is presented as a example, on the basis of which the influence of inhomogeneity of the elastic material on the magnitude of the stamp displacement parameters is investigated. An asymptotic analysis is performed for the solution in the case when the elastic characteristics of the material become rapidly oscillating functions.

Three-dimensional nonaxisymmetric problems of the theory of stability of composite materials with a metallic matrix

Three-dimensional nonaxisymmetric problems of the theory of stability of composite materials with a metallic matrix

Spatial nonaxisymmetric problems of the theory of stability of laminar highly elastic composite materials

Spatial nonaxisymmetric problems of the theory of stability of laminar highly elastic composite materials

INTEGRAL EQUATIONS OF THE SECOND KIND FOR STOKES FLOW: DIRECT SOLUTION FOR PHYSICAL VARIABLES AND REMOVAL OF INHERENT ACCURACY LIMITATIONS

Boundary integral methods offer the most attractive combination of generality and computational efficiency for a wide class of particulate Stokes flow problems. Integral equations of the first kind have been numerically applied for more than a decade, whereas those of the second kind are numerically better behaved but involve abstract nonphysical density distributions and have not gained much popularity in applications. We show how the latter may be used for the direct solution of mobility problems, and how the surface tractions corresponding to rigid body motion of a particle may be easily found, thus removing the major disadvantages of the second kind formulations. For the numerical examples we also show how Fourier analysis may be applied to non-axisymmetric problems with axisymmetric boundaries to yield one-dimensional Fredholm integral equations of the second kind. As an application we solve the resistance problem with a numerically efficient quadrature collocation method that avoids the complication...

Construction of the general solution of a nonaxisymmetric dynamical problem for a shallow spherical shell with a finite shear rigidity

Construction of the general solution of a nonaxisymmetric dynamical problem for a shallow spherical shell with a finite shear rigidity

Free and constrained inflation of elastic membranes in relation to thermoforming — non-axisymmetric problems

This paper deals with the inflation of elastic membranes of general shapes in the context of thermoforming of heated polymeric sheets against relatively cold moulding surfaces. A previous paper (1) has dealt with axisymmetric problems. As before, both theoretical and experimental investigations are reported. The theoretical part consists of a self-contained finite element formulation for large deformation, free and constrained inflation of isotropic elastic membranes. The computer program developed on the basis of presented formulation is applied to free and constrained inflation of plane elliptical membranes of aspect ratios 2 and 4. The material model adopted is that of a single constant neo-Hookean elastic material. The constrained analyses are carried out for inflation against: (a) cylindrical walls (90 degree elliptical cones), (b) 60 degree elliptical cones, and (c) horizontal plates. Separate analyses are performed by assuming the contact to be either slipless or frictionless. The theoretical results are compared with the experimental ones for free inflation and constrained inflation cases (a) and (c).