Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders

Exact solutions are obtained for transient torsio- nal responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free-free, free-fixed and fixed-fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.     

Upper and lower bounds for the optimum design of structures with random resistance

Summary The expected value of the minimum weight of material necessary for a structure in order to support assigned loads is considered, when the « resistance » of each element is a random quantity.By going back to a theorem which gives a lower bound for this expected value, a way to compute an upper bound is shown.A way of evaluating the variation of the interval of the existence of the average is then set out.Such bounds are functions of random variables but are evaluated on the basis of a deterministic limit design.

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Sommario Richiamato un teorema[4] che fornisce un minorante del valore medio del volume minimo di strutture soggette a carichi noti e dotate di elementi costituiti da materiale a resistenza aleatoria, si dimostra l'esistenza di un maggiorante.Si dimostra poi come sia possibile stabilire un maggiorante ed un minorante della varianza del minimo volume, e quindi come si possa apprezzare il rischio d'errore connesso alla aleatorietà delle resistenze, nella valutazione del volume minimo.I limiti indicati, benchè derivati da elaborazioni statistiche sulle resistenze, risultano da un progetto ottimale sviluppato con soli calcoli deterministici.

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Study supported by the C.N.R. (Consiglio Nazionale delle Ricerche).     

粘弹性厚壁筒问题的辛本征解方法

将哈密顿体系引进到粘弹性力学厚壁筒问题中,在辛体系下重新描述了基本问题,即建立了正则方程组。借助于积分变换,得到了拉伸、扭转和弯曲等问题的解以及有边界局部效应的解。将原问题归结为辛几何空间中的零本征值本征解和非零本征值本征解问题,从而建立了一种有效的分析问题方法和数值方法。为解决同类问题提供了一条可行的路径。     

Deformations of helical spring and cuboid into hollow cylinders

We study finite inhomogeneous deformations of a helical spring with a rectangular cross-section and a long cuboid. Two surfaces of the spring or the cuboid are joined to obtain a hollow cylinder. When body forces are absent the equilibrium equations reduce to ordinary differential equations. The stress-strain states are the same in each cross-section. The proposed deformations correspond to an inflation, an extension and a torsion of the obtained hollow cylinders. If the obtained cylinders are free of external applied loads, then they have residual stresses.     

The dynamic coupling of torsional and flexural strains in cylindrical helical springs

Summary We have examined, both theoretically and experimentally, the coupling between torsional and flexural strains in cylindrical helical springs axially excited. The analytical investigation, developed on the basis of the theory till now at our disposal, has shown the existence of two series of resonance conditions: the first one corresponds to the one already known for the axial vibrations, the other one is relative to the vibrations constituted by the rotations of the coils around the spring axis. The experimental research, developed into two springs having different characteristics and subjected to different assembly preloadings, has shown that the coupling between torsional and flexural strains is much more complex than the one indicated by theory, because of the presence of lateral deformations of the spring. The different tests, effected with an exciting variable frequency in a field below the second natural frequency of the axial vibrations, have shown the existence of more than two resonances.

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Sommario Viene esaminato, sia teoricamente che sperimentalmente, l'accoppiamento tra le deformazioni torsionali e flessionali in molle ad elica cilindrica sottoposte ad eccitazione secondo il proprio asse. L'indagine analitica, svolta sulla base della teoria finora disponibile, ha messo in evidenza l'esistenza di due serie di condizioni di risonanza: la prima corrisponde a quella ben nota per le vibrazioni assiali, la seconda è relativa alle vibrazioni costituite da rotazioni delle spire intorno all'asse della molla. L'indagine sperimentale, svolta su due molle aventi diverse caratteristiche e sottoposte a precarichi di montaggio diversi di volta in volta, ha mostrato che l'accoppiamento fra le deformazioni torsionali e flessionali è ben più complesso di quello indicato dalla teoria, a causa della presenza di deformazioni laterali della molla. Le diverse prove, effettuate con frequenza eccitante variabile in un campo al di sotto della seconda frequenza naturale delle vibrazioni assiali, hanno messo in evidenza l'esistenza di più di due risonanze.

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Research supported by the Italian Research Committee (C.N.R.).     

On bounds for the torsional stiffness of shafts of varying circular cross section

We state upper and lower bound formulas for the torsional stiffness of shafts of varying circular cross section, in accordance with the classical Michell formulation of this problem, through use of the principles of minimum potential and complementary energy. The general results are used to obtain explicit first-approximation bounds which, for the limiting case of the cylindrical shaft, reproduce the known elementary exact results. It is conjectured that the first-approximation lower bound is significantly closer to the exact result than the first-approximation upper bound.A report on work supported by the Office of Naval Research, Washington, D.C.     

Upper and lower bounds of buckling loads

Upper and lower bounds of buckling load for a nonuniform elastic column under conservative loading are considered. Compatible admissible moment and displacement functions are expressed in terms of a compatible coordinate system. The generalized Timoshenko Quotient and the modified Schreyer and Shih formula are the proposed upper and lower bounds. Both bounds when iterated converge to the exact buckling load. The method described here is simple and convenient and applies to all self-adjoint problems without exception.     

On 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders

In this paper, 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders have been carried out by Pickett's double series expansion method^[1]. Through expanding the displacement potentials as the sum of fourier series and Fourier-Bessel series, we could express the coefficients of one series by those of another under certain boundary conditions. Thus, a set of linear algebraic equations were derived. Solving these equations, we could obtain the solutions of the problems. Numerical examples have been given to show that the method presented here is workable for practical applications.     

The transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems

By virtue of the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of the displacements, stresses, electric displacements and electric potentials are finally obtained. The present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are also presented.     

Improved upper and lower bounds for deflections of orthotropic cantilever beams

Formulas for upper and lower deflection bounds, in terms of appropriately applied approximations to potential and complementary energy expressions, are evaluated on the basis of variational problems which involve fourth-order ordinary Euler differential equations, with associated Euler and constraint boundary conditions. The paper obtains new information on the order of magnitude of effects which modify the results of elementary beam theory through the influence of transverse shear and normal strain deformations, including the delineation of boundary layer effects, with one or two such layers, depending on the degree of orthotropy of the material of the beam.     

Influence of corrugation frequency and amplitude on the stress state of hollow elliptic cylinders

The influence of corrugation frequency and amplitude on the displacement and stress fields is analyzed based on an approach to solving stress problems for corrugated elliptical cylinders with certain end conditions.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 87–93, September 2004.     

Effect of the dilation loss on the vibration heating of hollow viscoelastic cylinders

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 8, pp. 68–73, August, 1988.     

Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders

The control of the plastic flow mechanism during axial collapse of metallic hollow cylinders is of particular interest in the present work for the absorbed energy. Hence, an experimental methodology is developed during which some different tubular structures are loaded under compressive quasi-static strain rate. These structures of various geometrical parameters η = R_m/t and λ = R_m/L (R_m: mean radius, L: initial length and t: thickness of tube) are made either from copper or aluminum considered as an energy dissipating system. At this point, the effects of both parameters on the mean collapse load and absorbed energy are appropriately studied. The role of η ratio, which has been largely investigated previously, is studied again. Moreover, it is found that the λ ratio has a non-negligible influence on the deformation mode for a given η. It is well known that the absorbed energy is influenced by the deformation mechanism, i.e., for the axisymmetric mode, the related absorbed energy becomes more important than that of the diamond fold mechanism for a given cylinder. Accordingly, to maximize the absorbed energy, two different structural solutions, namely fixed-ends and subdivided structure, are developed for encouraging the axisymmetric mode. It is convenient to consider the classical axial collapse situation (noted as free-ends) as a comparison reference. In this work, it is recognized that the subdivided solution is relatively the best solution. As a result, the absorbed energy increases up to 21% in comparison with the free-ends situation for copper tubes.     

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.     

Quadratic nonlinear torsional hyperelastic waves in isotropic cylinders: Primary analysis of evolution

Nonlinear torsional waves propagating along an infinite isotropic cylinder are considered. The hyperelasticity of the cylinder is described by the Murnaghan potential. An approximate analytical solution of the problem is obtained. The evolution of the initial waveprofile in cylinders made of composite materials reinforced with microscale particles is simulated numerically. It is shown that the approximation used to describe the variation in the amplitude inward the cylinder is quite accurate. Three-dimensional plots of evolution are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 75–85, March 2008.     

Elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders by analytical method

An elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders subjected to uniformly-distributed dynamic pressures at boundary surfaces is presented. The material properties, except Poisson’s ratio, are assumed to vary through the thickness according to a power law function. To achieve an exact solution, the dynamic radial displacement is divided into two quasi-static and dynamic parts, and for each part, an analytical solution is derived. The quasi-static solution is obtained by means of Euler’s equation, and the dynamic solution is derived using the method of the separation of variables and the orthogonal expansion technique. The radial displacement and stress distributions are plotted for various functionally graded material (FGM) hollow cylinders under different dynamic loads, and the advantages of the presented method are discussed. The proposed analytical solution is suitable for analyzing various arrangements of hollow FGM cylinders with arbitrary thickness and arbitrary initial conditions, which are subjected to arbitrary forms of dynamic pressures distributed uniformly on their boundary surfaces.     

Elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders by analytical method

An elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders subjected to uniformly-distributed dynamic pressures at boundary surfaces is presented. The material properties, except Poisson’s ratio, are assumed to vary through the thickness according to a power law function. To achieve an exact solution, the dynamic radial displacement is divided into two quasi-static and dynamic parts, and for each part, an analytical solution is derived. The quasi-static solution is obtained by means of Euler’s equation, and the dynamic solution is derived using the method of the separation of variables and the orthogonal expansion technique. The radial displacement and stress distributions are plotted for various functionally graded material (FGM) hollow cylinders under different dynamic loads, and the advantages of the presented method are discussed. The proposed analytical solution is suitable for analyzing various arrangements of hollow FGM cylinders with arbitrary thickness and arbitrary initial conditions, which are subjected to arbitrary forms of dynamic pressures distributed uniformly on their boundary surfaces.