Circumferential Coordinate Research Articles

Analysis of wave dispersion in cylindrical shells with anisotropic layers by Rayleigh–Ritz expansions

Recent advances in composite materials offer the potential to dramatically affect the dispersions of helical waves that propagate in layered cylindrical shells. Effective choices and configurations of these materials require rapid dispersion predictions that take into account general anisotropy and variations of elastic variables through the thickness of each layer. This presentation describes a solution to this problem that is based on a previously developed theory [J. G. McDaniel and J. H. Ginsberg, J. Appl. Mech. 60, 463–469 (1993)] for the vibrations of cylindrical shells with isotropic layers. The approach uses propagating wave representations in the axial and circumferential coordinates. A series expansion with polynomial basis functions represents dependences on the radial coordinate and equations for the series coefficients are derived using the Rayleigh–Ritz method. These equations yield a dispersion relation that is solved for the complex-valued axial wavenumber at each frequency and circumferential harmonic. Examples illustrate the interface of this approach with optimization algorithms that identify layer designs possessing specified dispersion properties. [Work supported by ONR.]

Anisotropic Elastic Tubes of Arbitrary Cross Section Under Arbitrary End Loads: Separation of Beamlike and Decaying Solutions

Abstract First approximation analytical solutions are constructed for finite and semi-infinite, fully anisotropic elastic tubes of constant thickness h and arbitrary cross section, subject to purely kinetic or purely kinematic boundary conditions. Final results contain relative errors of O(h∕R), where R is some equivalent cross sectional radius. Solutions are decomposed into the sum of an exact beamlike or Saint-Venant solution, treated in Ladevèze et al. (Int. J. Solids Struct., 41, pp. 1925–1944, 2004) and extended in an appendix; a rapidly decaying edge-zone solution; and a slowly decaying semi-membrane-inextensional-bending (MB) solution. Explicit conditions on the boundary data are given that guarantee decaying solutions. The MB solutions are expressed as an infinite series of complex-valued exponential functions times real-valued one-dimensional eigenfunctions which satisfy a fourth-order differential equation in the circumferential coordinate and depend on the pointwise cross sectional curvature only.

Aerodynamic design method of cascade profiles based on load and blade thickness distribution

A cascade profile design method was proposed using the aerodynamic load and blade thickness distribution as the design constraints, which were correspondent to the demands from the aerodynamic characteristics and the blade strength. These constraints, together with all the other boundary conditions, were involved in the stationary conditions of a variational principle, in which the angle-function was employed as the unknown function. The angle-function (i. e., the circumferential angular coordinate) was defined in the image plane composed of the stream function coordinate (circumferential direction) and streamline coordinate. The solution domain, i. e., the blade-to-blade passage, was transformed into a square in the image plane, while the blade contour was projected to a straight line; thus, the difficulty of the unknown blade geometry was avoided. The finite element method was employed to establish the calculation code. Applications show that this method can satisfy the design requests on the blade profile from both aerodynamic and strength respects. In addition, quite different from the most inverse-problem approaches that often encounter difficulties in the convergence of iteration, the present method shows a stable and fast convergence tendency. This will be significant for engineering applications.

State space solution of non-axisymmetric Biot consolidation problem for multilayered porous media

In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements at layer interfaces as basic unknown functions, two sets of partial differential equations, which are independent each other, are formulated. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. Transfer matrices describing the transfer relation between the state vectors for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, three cases are studied for the lower surface: (1) permeable rough rigid base, (2) impermeable rough rigid base, and (3) poroelastic half space. The explicit solution in the transform space is presented. Considering the continuity condition at layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for multilayered semi-infinite porous media are presented in the integral form. The time histories of displacements, stresses and pore pressure are obtained by solving a linear equation system for discrete values of Laplace–Hankel transform inversions.

Three-Dimensional Vibration Analysis of Paraboloidal Shells.

A method of analysis is presented for determining the free vibration frequencies and mode shapes of open paraboloidal shells of revolution having arbitrary thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell, as well as the inner and outer curved surfaces, may be free or may be subjected to any degree of constraint. The strain energy of deformation, as well as the kinetic energy of motion, are formulated in terms of three displacement components which are tangent or normal to the shell middle surface. The displacements are taken as periodic in the circumferential coordinate and in time, and as polynomials of arbitrary degree in the other two coordinates, and the Ritz method is used to formulate the eigenvalue problem. Convergence studies are presented, and frequencies are given for moderately thick and thick, moderately deep and deep, paraboloidal shells of uniform and variable thickness.

Effect of heat flow on heat‐transfer characteristics for a supersonic spatial flow around a spherically blunted cone

Heat‐transfer processes for a supersonic spatial flow around a spherically blunted cone were studied by solving direct and inverse three‐dimensional problems taking into account heat flow along the longitudinal and circumferential coordinates. It is shown that highly heat‐conducting materials can be used to advantage to decrease the maximum temperatures on the windward side of streamline bodies.

Stability of Orthotropic Corrugated Cylindrical Shells under Axial Compression

The problem of stability of orthotropic noncircular cylindrical shells whose cross section can be described by a function in the form of superposition of a constant and a multiperiodic cosinusoid is considered. The solution of this problem is based on a rigorous consideration of the shell geometry and on the representation of the resolving functions in terms of trigonometric series in the circumferential coordinate. The determination of the bifurcation load is reduced to finding the minimum eigenvalue of a sequence of infinite systems of homogeneous algebraic equations. The effect of corrugation on the critical load of thin glass- and boron-reinforced shells of arbitrary length is analyzed. The data on the efficiency of corrugated shells, in comparison with circular ones, in relation to the mechanical properties of materials are obtained. The accuracy of the calculation procedure is estimated in the case where the corrugated shell is simulated by an equivalent circular one.

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heat‐transfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heat‐conducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one‐, two‐, and three‐dimensional formulations for different shell materials are compared. The error of the thin‐wall method in determining the heat flux on the heat‐loaded boundary of the body is estimated.

Calculation of the thermoelastoplastic nonaxisymmetric stress-strain state of layered orthotropic shells of revolution

The methods for determining the nonaxisymmetric thermoelastoplastic stress-strain state of layered orthotropic shells of revolution are developed. It is assumed that the layered package deforms without mutual slippage or separation of layers. The problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. In the isotropic layers, plastic deformations may appear, whereas the orthotropic layers deform in the elastic region. It is assumed that the mechanical properties of the materials are temperature-dependent. The thermoplasticity equations are presented in a form corresponding to the method of additional deformations. The order of the system of partial differential equations obtained is reduced with the help of trigonometric series in the circumferential coordinate. The resulting systems of ordinary differential equations are solved by the Godunov technique of discrete orthogonalization. The nonaxisymmetric thermoelastoplastic stress-strain states of layered shells of revolution are considered as examples.

Response and Failure Characteristics of Internally Pressurized Elliptical Composite Cylinders

The characteristics of the response of internally pressurized thin-walled elliptical composite cylinders, including failure, are discussed. The influence of the elliptical geometry on response is illustrated by comparison with a circular cylinder. It is shown that with internal pressure elliptical cylinders tend to become more circular by exhibiting inward normal displacements at some circumferential locations and by the existence of circumferential displacements. The influence of material orthotropy is illustrated by considering axially stiff, circumferentially stiff, and quasi-isotropic laminates. The results show that orthotropy can he used to counter the influence of the circumferentially varying curvature by producing circumferential strains at the midcylinder axial location of a circumferentially stiff cylinder that are almost independent of the circumferential coordinate, much like the axisymmetric response of a circular cylinder. The influence of geometric nonlinearities is studied by inclusion of the von Karman terms in the strain-displacement relations, and it is shown that one of the primary effects is flattening of the ellipse in the regions near the ends of the minor diameter. Because of varying curvature in the circumferential direction, there can be, effectively, a stress concentration at certain circumferential locations, which are sites of failure initiation, the exact location depending on material orthoropy. To study failure initiation, two failure criteria are considered: the interactive Hashin criterion and the noninteractive maximum stress criterion. These criteria are used to compute the pressures to cause first matrix cracking and first fiber failure. It is demonstrated that the predictions of the two criteria are not that different, and the pressure to cause first fiber failure is about twice the pressure to cause first matrix failure. Additionally, it is shown that matrix cracking begins in the circumferentially stiff cylinder at lower pressures than in the axially stiff and quasi-isotropic cylinders, and the location of failure is different.

Analysis of Vibration Characteristics of a Rotating Ring with the Inclined Rotation Axis.

The vibrational characteristics of a rotating ring with the inclined rotation axis are examined. The fundamental equations are derived depending on the Hamilton's principle, and the finite strain is used to take into consideration the product terms of steady-state displacements due to rotation and vibrational displacements, in addition to the centrifugal and Corioris forces. The both displacements are expressed by the Fourier series with respect to the circumferential coordinate, and by using orthogonality relations of trigonometric functions, the vibration equations are obtained. The circumferential wave number n are taken from 0 to 2 after confirming the convergence of solution up to n=2. The characteristic roots of those equations are calculated for the inclined angles 0°, 45° and 90°. For the case of the inclined angle 0° and 90°, all the vibrational modes are stable, whereas several modes which are rigid-body modes in the case of no-rotation become unstable for the case of inclined angle 45°.

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.

Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness

The free vibration analysis of laminated conical shells with variable stiffness is presented using the method of differential quadrature (DQ). The stiffness coefficients are assumed to be functions of the circumferential coordinate that may be more close to the realistic applications. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations. In the DQ method, the governing equations and the corresponding boundary conditions are replaced by a system of simultaneously algebraic equations in terms of the function values of all the sampling points in the whole domain. These equations constitute a well-posed eigenvalue problem where the total number of equations is identical to that of unknowns and they can be solved readily. By vanishing the semivertex angle ( α) of the conical shell, we can reduce the formulation of laminated conical shells to that of laminated cylindrical shells of which stiffness coefficients are the constants. Besides, the present formulation is also applicable to the analysis of annular plates by letting α= π/2. Illustrative examples are given to demonstrate the performance of the present DQ method for the analysis of various structures (annular plates, cylindrical shells and conical shells). The discrepancies between the analyses of laminated conical shells considering the constant stiffness and the variable stiffness are mainly concerned.

Solutions of Complete Circular Cylindrical Shell under Concentrated Loads

Series solutions to the differential equations for a complete cylindrical shell under various point loads are developed. Instead of expanding the solution series in both axial and circumferential directions, the solution obtained here is only expanded in circumferential coordinates. Each term in the series represents an exact solution to the set of basic equations under a decomposed ring loading.

Thermoelastoplastic nonaxisymmetric stress-strain analysis of laminated shells of revolution

A technique for nonaxisymmetric thermoelastoplastic stress–strain analysis of laminated shells of revolution is developed. It is assumed that there is no slippage and the layers are not separated. The problem is solved using the geometrically linear theory of shells based on the Kirchhoff–Love hypotheses. The thermoplastic relations are written down in the form of the method of elastic solutions. The order of the system of partial differential equations obtained is reduced by means of trigonometric series in the circumferential coordinate. The systems of ordinary differential equations thus obtained are solved by Godunov's discrete-orthogonalization method. The nonaxisymmetric thermoelastoplastic stress–strain state of a two-layered shell is analyzed as an example

Elastic Equilibrium of Circumferentially Inhomogeneous Orthotropic Cylindrical Shells of Arbitrary Thickness

An approach is proposed to determine the three-dimensional stress–strain state of hollow orthotropic cylinders with elastic characteristics variable in the circumferential direction. The solution of the problem is represented as a double Fourier series in the axial and circumferential coordinates. A resolving system of ordinary differential equations of high order is derived. An analysis is made of the stress state of a cylinder made of a composite with reinforcement density variable in the circumferential direction

Application of the Spectral Technique to the Analysis of Dynamically Loaded Journal Bearings

Successful use of a realistic bearing model in engineering practice depends heavily on the efficiency of the tool used. The analysis of dynamically loaded bearings requires repeated solutions of the Reynolds equation at each point of the journal orbit, therefore, the method used for this problem should be very fast. A new (spectral) method is presented, which allows the Reynolds equation to be solved sufficiently quickly and accurately. This method can be defined as a modification of the Galerkin method for partial differential equations as suggested by Kantorovich. The solution of the Reynolds equation is sought as a Fourier series with respect to the circumferential coordinate. This allows the original partial differential equation to be reduced to a system of ordinary differential equations which, in turn, is further approximated with a linear algebraic system. Since the matrix of this system is banded, the system can be solved efficiently with simple recursive relations. A novel method is introduced to determine the loaded zone corresponding to the Reynolds boundary conditions. The method uses ideas of functional analysis. Consideration of the loaded zone as a function, along with a condition for the pressure to be non-negative, allows a functional equation to be written and then solved with the Euler or Runge-Kutta methods. A comparison with published FE results for a steady-state case as well as for the dynamic load is shown.

Natural frequencies of nonaxisymmetric vibration of prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle. The thin shell theory used in this derivation includes three displacements and two changes of curvature. The effects of membrane, bending, shear deformations, and rotatory inertias are included in this theory. The resulting five partial differential equations are self-adjoint and positive definite. The nonaxisymmetric modal solutions are expanded in a doubly infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate, and circular functions of the circumferential coordinate. The natural frequencies and the mode shapes were obtained by the Galerkin method for each circumferential mode. Numerical results were obtained for several shell thickness-to-length ratios ranging from 0.005 to 0.1, and for various diameter-to-length ratios, including the limiting case of a spherical shell. [Work supported by Office of Naval Research and the Navy/ASEE Summer Faculty Program.]

Influence of Inlet Conditions on the Thermohydrodynamic State of a Fully Circumferentially Grooved Journal Bearing

A thermohydrodynamic (THD) analysis of a fully circumferentially grooved hydrodynamic bearing is presented. The pressure distribution is obtained using the short bearing approximation taking into account the viscosity variation in the radial and circumferential coordinates. The axial temperature variation is also included by an axial averaging technique, which incorporates the supply pressure and film entry temperature in the energy equation. It is found that the determination of the lubricant temperature at the entry to the film plays an important role in the overall temperature distribution in the bearing. A simplified approach for determining this temperature is presented. An extensive set of experimental results performed by Maki and Ezzat (1980, ASME J. Lubr. Technol., 102, pp. 8–14) is used for validation purposes. The results show that mixing in the inlet groove may cause the film entry temperature to be significantly different from the nominal supply temperature and hence have a significant influence on the bearing temperature.

Stability problem of a circular sandwich ring under uniform external pressure

The solution of the stability problem of a circular sandwich ring under uniform external pressure is given in a refined statement. The need to determinate the precritical stresses in load-bearing layers in the refined statement with regard to the transverse compression of the core is established, which is the basis for the detection of the mixed flexural buckling forms (BFs) with more than two half-waves along the circumferential coordinate (n>2). It is found that sandwich structures with a determining parameter of transverse compression corresponding to the limit of transition from the mixed BFs to synphasic ones are the most efficient from the weight viewpoint.