Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks

This paper provides free vibration data for cylindrical elastic solids, specifically thick circular plates and cylinders with V-notches and sharp radial cracks, for which no extensive previously published database is known to exist. Bending moment and shear force singularities are known to exist at the sharp reentrant corner of a thick V-notched plate under transverse vibratory motion, and three-dimensional (3-D) normal and transverse shear stresses are known to exist at the sharp reentrant terminus edge of a V-notched cylindrical elastic solid under 3-D free vibration. A theoretical analysis is done in this work utilizing a variational Ritz procedure including these essential singularity effects. The procedure incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of “edge functions” that explicitly model the 3-D stress singularities which exist along a reentrant terminus edge (i.e., α>180°) of the V-notch. The first set of polynomials guarantees convergence to exact frequencies, as sufficient terms are retained. The second set of edge functions—in addition to representing the corner stress singularities—substantially accelerates the convergence of frequency solutions. This is demonstrated through extensive convergence studies that have been carried out by the investigators. Numerical analysis has been carried out and the results have been given for cylindrical elastic solids with various V-notch angles and depths. The relative depth of the V-notch is defined as (1−c/a), and the notch angle is defined as (360°−α). For a very small notch angle (1° or less), the notch may be regarded as a “sharp radial crack.” Accurate (four significant figure) frequencies are presented for a wide spectrum of notch angles (360°−α), depths (1−c/a), and thickness ratios (a/h for plates and h/a for cylinders). An extended database of frequencies for completely free thick sectorial, semi-circular, and segmented plates and cylinders are also reported herein as interesting special cases. A generalization of the elasticity-based Ritz analysis and findings applicable here is an arbitrarily shaped V-notched cylindrical solid, being a surface traced out by a family of generatrix, which pass through the circumference of an arbitrarily shaped V-notched directrix curve, r(θ), several of which are described for future investigations and close extensions of this work.     

A sector finite element for dynamic analysis of thick plates

A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissner's thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.     

Free vibration of annular sector plates with edges possessing different degrees of rotational restraint

Results for the natural frequencies of annular sector plates possessing different degrees of elastic restraint along the edges are presented. The analysis is based on a numerical method developed by the author. The functions in the circumferential direction satisfying the boundary conditions along the radial edges, which are required in the analysis, are indicated. To the best of the author's knowledge, no previous results exist for such plates.     

圆薄板大撓度问题

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.     

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.     

Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

EXACT BUCKLING AND VIBRATION SOLUTIONS FOR STEPPED RECTANGULAR PLATES

This paper is concerned with the determination of exact buckling loads and vibration frequencies of multi-stepped rectangular plates based on the classical thin (Kirchhoff) plate theory. The plate is assumed to have two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The proposed analytical method for solution involves the Levy method and the state-space technique. By using this analytical method, exact buckling and vibration solutions are obtained for rectangular plates having one- and two-step thickness variations. These exact solutions are extremely useful as benchmark values for researchers developing numerical techniques and software for analyzing non-uniform thickness plates.     

Experimental investigation of free vibrations of clamped sector plates

The real time technique of time-averaged holographic interferometry has been applied to determine the natural frequencies and the corresponding mode shapes for the transverse vibrations of clamped wedge-shaped and ring-shaped sector plates. Over 200 resonant modes have been obtained for wedge-shaped sector plates and over 170 for ring-shaped sector plates. The natural frequencies obtained have been expressed in terms of a dimensionless frequency parameter, and the results are shown graphically as a function of the sector angle for the wedge-shaped plates and of the radii ratio for the ring-shaped sector plates, respectively. Some of the present results for wedge-shaped plates are compared with the analytical values obtained by other authors.     

On computation of random response of a clamped plate

The free vibrations of elastically connected circular plate systems with elastically restrained edges and initial radial tensions are investigated analytically. By using the equations developed for the general n-plate system, the plate systems consisting of three and two identical plates with identical boundary conditions and a uniform radial tension are treated in detail. Both axisymmetric and non-axisymmetric vibrations are considered. Attention is directed to the influence of the radial tension and the elastic edge constraints on the first nine eigenvalues and the corresponding natural frequencies of the systems.     

Vibration of plates with straight-line,clamped and free edges

This paper deals with vibration problems of thin plates having straight-line, mutually perpendicular, clamped and free edges and subjected to a load consisting of a set of transverse, arbitrarily located random forces. It is assumed that the number of edges of a plate forming recurring figures is optional but each of these edges is either clamped or free along its entire length. The procedure for solving the boundary problem based upon the R-functions method and for estimation of transverse displacements based upon the correlation analysis is presented. Numerical calculations are carried out for two example plates.     

Resonance frequencies and mode shapes of elastically restrained,prestressed annular plates attached together by flexible cores

The free vibrations of annular plates attached together by flexible cores are studied analytically. Both axisymmetric and non-axisymmetric vibrations are considered. The plates are elastically constrained against rotation at the inner and outer edges. At the same time, the plates are subjected to initial radial tensions. Detailed analysis is worked out for systems consisting of five through two identical plates with identical boundary conditions and a uniform radial tension. General frequency equations and mode shapes are developed. The first nine eigenvalues are calculated for a plate system having identically constrained inside and outside edges and are tabulated as functions of the initial tension parameter, the elastic edge constraint parameter and the ratio of inner to outer radius. The orthogonality property of the mode function is also discussed.     

Forced axisymmetric response of circular plates

Forced axisymmetric response of circular plates is considered on the basis of an improved theory which takes into account the effect of rotatory inertia and shear deformation. Explicit solutions are obtained for distributed, ring and concentrated impulsive loadings with half-sine, triangular, blast and rectangular pulse shapes. Natural frequencies are presented for the first 20 modes of four plate models with clamped and supported edges. Numerical results are presented to illustrate the differences between the improved and classical theories, and the effects of edge conditions, loading conditions, pulse shapes and pulse durations.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

Aerosol Synthesis of Fullerene Nanocrystals in Controlled Flow Reactor Conditions

Fullerene nanocrystals in the size range 30–300 nm were produced starting from atomized droplets of C₆₀ in toluene. The experiments were carried out under well-controlled conditions in a laminar flow reactor at temperatures of 20–600°C. Particle transformation and crystallization mechanisms of polydisperse and monodisperse (size classified) fullerene aerosol particles were studied. The results show that fullerene particles are roughly spherical having pores and voids at temperatures of 300°C and below. Particles are already crystalline and likely fine-grained at 20°C and they are polycrystalline at temperatures up to 300°C. At 400°C monodisperse particles evaporate almost completely due to their low mass concentration. Polydisperse particles are crystalline, but sometimes heavily faulted. At 500°C most of the particles are clearly faceted. In certain conditions, almost all particles are hexagonal platelets having planar defects parallel to large (111) faces. We suggest that at 500°C fullerene particles are partially vaporized forming residuals with lamellar defects such as twins and stacking faults, which promote crystal growth during synthesis. Subsequently fullerene vapor is condensed on faces with defects and hexagonal particles are grown by a re-entrant corner growth mechanism. At 600°C particles are single crystals, but they have a less distinct shape due to higher vaporization of fullerene. The final size and shape of the particles are mainly determined at the reactor outlet in the short time when the aerosol cools.     

Free vibration of annular sector plates by an integral equation technique

A numerical method is presented for the free vibration analysis of polar orthotropic clamped annular sector plates. The results are compared with analytical and experimental values of other investigators. A parametric study has been done by varying the sector angle and radii ratio. The frequencies for isotropic and orthotropic cases are presented in the form of graphs.     

Free vibration analysis of doubly curved shallow shells using the Superposition-Galerkin method

This paper demonstrates the applicability of the Superposition Method for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of simply supported and clamped edges. The same building block yields the natural frequencies for 55 combinations of edge conditions. The natural frequency parameters of the shells were obtained using the Superposition-Galerkin Method (SGM) for seven sets of boundary conditions, several different curvature ratios and two aspect ratios. The SGM uses approximate steady state solutions as building blocks but the method proves to be accurate and efficient. It has also been shown that even with approximate building blocks, the monotonic nature of convergence of the natural frequencies with respect to the number of driving coefficients holds, as long as the number of admissible functions in the steady state solution is kept constant. The results for natural frequencies of the seven boundary conditions may be considered as benchmarks.     

Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.     

Out-of-plane free vibrations of curved beams with variable curvature

The differential equations governing out-of-plane free vibrations of the elastic, horizontally curved beams with variable curvature are derived and solved numerically to obtain natural frequencies and mode shapes for parabolic, sinusoidal and elliptic beams with hinged–hinged, hinged–clamped, and clamped–clamped end constraints, in which the effects of the rotatory and torsional inertias and shear deformation are included. Experimental measures of frequencies for several laboratory-scale parabolic models serve to validate the theoretical results.     

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.     

Sharp corners as sources of spiral pairs

It is demonstrated that using the FitzHugh-Nagumo model, stimulation of excitable media inside a region possessing sharp corners, can lead to the appearance of sources of spiral-pairs of sustained activity. The two conditions for such source creation are: The corners should be less than 120° and the range of stimulating amplitudes should be small, occurring just above the threshold value and decreasing with the corner angle. The basic mechanisms driving the phenomenon are discussed. These include: A. If the corner angle is below 120°, the wave generated inside cannot emerge at the corner tip, resulting in the creation of two free edges which start spiraling towards each other. B. Spiraling must be strong enough; otherwise annihilation of the rotating arms would occur too soon to create a viable source. C. The intricacies of the different radii involved are elucidated. Possible applications in heart stimulation and in chemical reactions are considered.