Influence of the initial imperfection on the non-linear buckling response of FGM truncated conical shells

Non-linear buckling analyses of imperfect functionally graded truncated conical shells with simply supported boundary conditions and subjected to an axial compressive load have been presented in this work. The material properties of functionally graded shells are assumed to vary continuously through the thickness of the shell. The non-linear prebuckling deformations and initial geometric imperfections of an FGM truncated conical shell are both taken into account. The fundamental relations, modified Donnell type non-linear stability and compatibility equations of an imperfect FGM truncated conical shell are obtained and are solved by superposition and Galerkin methods, and the upper and lower critical axial loads has been found analytically. The numerical illustrations concern the non-linear buckling response of FGM truncated conical shells with different values of truncated conical shell parameters, initial imperfections and compositional profiles. Comparing the results of this study with those in the literature validates the present analysis.     

Free vibration of composite laminated conical shells

Using a particularly convenient coordinate system, a simple and exact solution is obtained directly for the Donnell-type governing equations of the free vibration of composite laminated conical shells, with orthotropic stretching-bending coupling. The solution is in the form of a power series, and its convergence condition is investigated. Numerical results, presented for the frequency parameters and the associated vibration wavenumbers of a series of conical shells under various boundary conditions and with different material coefficients, confirm the validity of the solution presented.     

Torsion and transverse sensing of conical shells

Conical shells are widely used as payload/rocket adapters in rocket fairing systems. Generally, the conical shells are clamped at the major end and free at the minor end, where the payload is mounted. This study focuses on the dynamic sensing of conical shells with fix-free boundary conditions (BCs) by using distributed piezoelectric helical sensors. Two types of motion are studied, i.e., the transverse modes and the torsion modes. The shear-type sensors for shells sensing are presented first. Formulations of sensing signals of a general shell of revolution are presented, and then simplified to conical shells. For sensing of transverse vibrations, thin piezoelectric sensors are laminated on the top surface. Two types of sensor distribution are considered: a fully distributed and a helical or diagonal laminated. The total signal consists of four components resulting from the four strain components, and each of them is evaluated in detail. For sensing of torsion vibrations, a meridional polarized shear-type sensor with side electrodes is layered on the top surface of the shell structure. Sensing signals of natural shell modes are also evaluated. Analyses show that, in low order modes, the sensing signals induced by the circumferential membrane strains are the primary components of the total signal generations. The numerical results indicate the optimal location of the sensors. The proposed method is capable of determining the modal participation factors, while the testing signal is available; it is also capable of determining the mode shapes by using several distributed sensor segments.     

Vibration of twisted laminated composite conical shells

A methodology for free vibration of a laminated composite conical shell with twist is proposed, in which a strain–displacement relationship of a twisted conical shell is given by considering the Green strain tensor on the general thin shell theory, the principle of virtual work is utilized, and the governing equation is formulated by the Rayleigh–Ritz procedure with algebraic polynomials in two elements as admissible displacement functions. The convergence, the accuracy and the validity of the methodology are verified by comparisons. As a result of the vibration frequencies and mode shapes, the effects of the laminated constructional and the geometric parameters, such as the number of laminae, the fiber orientation angles, the twist angle, the subtended angle and the taper ratio, on the vibration characteristics are studied by the present methodology.     

Plastic buckling of conical shells under axial compression

Analysis and experimental results are presented on the plastic axisymmetric buckling of steep, truncated conical shells under axial compression. Specimens of 6061-T6 aluminum and type 416 stainless steel were tested; in spite of the considerable difference in the stress-strain curves for the two materials, the buckling modes observed in the experiments for geometrically identical cones were the same. Perturbation analysis, which takes account of the continuous change in direction of the plastic strain rate vector during buckling, is found to describe the essential features of the observed buckling deformation.     

Pulse transmission in elastic exponential and conical shells

A theoretical and experimental investigation was conducted to ascertain the response of an axisymmetric exponential shell and of a hollow cone with identical terminal dimensions and lengths, both composed of aluminum, to central longitudinal impact by steel spheres. The principal tests were concerned with loading at the small end and an open distal section; in a few cases, the large end was covered by a relatively thin aluminum plate and subjected to central impact by the various strikers. The range of initial velocities for the 3.175, 6.35 and 12.7 mm diameter pneumatically-fired steel spheres was from 7.6 to 60 m/s, while 23.7 and 50.8 mm steel balls were propelled in a pendulum arrangement at about 1.2 m/s. The input was monitored by a quartz crystal, while the waves were detected by a set of foil gages. A one- and a two-dimensional numerical analysis were executed for a typical set of experimental conditions of the first wave transit in both systems when struck at the small end: the results were in good correspondence with experimental data. The exponential shell exhibited the theoretically predicted cut-off behavior beyond a certain pulse duration which was also found previously in the case of a geometrically similar solid exponential rod. Correspondingly, the conical shell evidenced enormous attenuation, but not complete cut-off beyond such critical impact duration when the wave process was initiated at the small end. The much greater dispersion and attenuation relative to that of the solid bars is primarily due to wall bending of the specimens.     

On the inextensional axial collapse of thin PVC conical shells

Inextensional collapse mechanisms are presented for the axial crumpling of thin-walled circular cones and frusta (truncated circular cones). Shortening of the (thin) shell height is achieved by folding in a non-symmetric diamond mode about stationary circumferential and inclined plastic hinges; collapse proceeds progressively from the narrower end of the conical shell during the passage of a travelling hinge. Expressions for the various mean crushing loads, when collapsing frusta of rigid-perfectly-plastic material, are developed. Theoretical collapse modes and predicted loads are compared with those obtained experimentally by collapsing rigid PVC conical shells of constant axial length, of various wall-thicknesses and semi-apical angles, as well as metal (aluminium alloy and low-carbon steel) conical shells of similar geometry; agreement is found to be good.     

An efficient approach for free vibration analysis of conical shells

In this paper, the global method of generalized differential quadrature (GDQ) is applied for the first time to study the free vibration of isotropic conical shells. The shell equations used are Love-type. The displacement fields are expressed as product of unknown functions along the axial direction and Fourier functions along the circumferential direction. The derivatives in both the governing equations and the boundary conditions are discretized by the GDQ method. Using the GDQ method, the natural frequencies can be easily and accurately obtained by using a considerably small number of grid points. The accuracy and efficiency of the GDQ method is examined by comparing the results with those in the literature and very good agreement is observed. The fundamental frequency parameters for four sets of boundary conditions and various semivertex angles are also shown in the paper.     

Neural potentials and micro-signals of non-linear deep and shallow conical shells

Conventional sensors, such as proximeters and accelerometers, are add-on devices usually adding additional weights to structures and machines. Health monitoring of flexible structures by electroactive smart materials has been investigated over the years. Thin-film piezoelectric material, e.g. polyvinylidene fluoride (PVDF) polymeric material, is a lightweight and dynamic sensitive material appearing to be a perfect candidate in monitoring structure's dynamic state and health status of flexible shell structures with complex geometries. The complexity of shell structures has thwarted the progress in studying the distributed sensing of shell structures. Linear distributed sensing of various structures have been studied, e.g. beams, plates, cylindrical shells, conical shells, spherical shells, paraboloidal shells and toroidal shells. However, distributed microscopic neural signals of non-linear shell structures has not been carried out rigorously. This study is to evaluate microscopic signals, modal voltages and distributed micro-neural signal components of truncated non-linear conical shells laminated with distributed infinitesimal piezoelectric neurons. Signal generation of distributed neuron sensors laminated on conical shells is defined first. The dynamic neural signal of truncated non-linear conical shells consists of microscopic linear and non-linear membrane components and linear bending component based on the von Karman geometric non-linearity. Micro-signals, modal voltages and distributed neural signal components of two different truncated non-linear conical shells are investigated and their sensitivities discussed.     

Influence of initial pressure on frequency characteristics of a rotating truncated circular conical shell

As a new global numerical approximate technique, the generalized differential quadrature (GDQ) method is used in this paper to study the influence of initial pressure load on the free vibration of a rotating thin isotropic truncated circular conical shell. The present motion governing equations include the influence of initial stress field due to the initial uniform pressure. The effects of initial hoop tension and the centrifugal and coriolis accelerations due to rotation are also considered. The influence of initial pressure on the frequency characteristics of the rotating conical shell is discussed in detail. For the examination of present work, frequency numerical comparisons are made with those available in published works, and very good agreement is achieved.     

Analysis of nonlinear vibration response of a functionally graded truncated conical shell with piezoelectric layers

The nonlinear vibration response of a functionally graded materials (FGMs) truncated conical shell with piezoelectric layers is analyzed. The vibration amplitude is suppressed by the positive and inverse piezoelectric effects. And the bifurcation phenomenon is described to reveal the motion state of the conical shell. Firstly, a truncated conical shell composed of three layers is described. And the effective material properties of the FG layer are defined by the Voigt model and the power law distribution. Next, the electric potentials of piezoelectric layers are defined as cosine distribution along the thickness direction. Meanwhile, the constant gain negative velocity feedback algorithm is used to suppress the vibration amplitude by the electric potential produced by the sensor layer. Thereafter, considering the first-order shear deformation theory and the von Karman nonlinearity, the relationship between the strain and displacement is defined. And the corresponding energy of the conical shell is calculated. After that, the motion equations of the conical shell are derived based on the Hamilton principle. Again, the nonlinear single degree of freedom equation is derived by the Galerkin method and the static condensation method. In the end, the nonlinear vibration response of FGMs truncated conical shell with piezoelectric layers under the external excitation is analyzed via using the harmonic balance method and the Runge-Kutta method. The effects of various parameters, such as ceramic volume fraction exponent, external excitation’s amplitude, control gain and geometric parameters on the nonlinear vibration response of the system are evaluated by case studies. Results indicate that the control gain plays an important role on the suppression of the vibration amplitude. The ceramic volume fraction exponents are not sensitive to the nonlinear vibration response compared with other parameters. The bifurcation behavior is observed under different parameters. The FGMs truncated conical shell with piezoelectric layers has three types of motion state, such as periodic motion, multi-periodic motion, and chaos motion.

     

Contrast between elastic and plastic buckling of axially compressed conical shells with edge constraint

Analysis of plastic buckling of axially compressed truncated conical shells taking account of edge constraint due to friction is presented and related to earlier experimental results. It is found that the time-dependence of the external load in the equilibrium equations is essential in the mathematical formulation. Localization of buckling deformation near the boundary appears to be a distinctive feature of plastic buckling, in contrast to elastic buckling treated on the basis of the same equilibrium equations, boundary conditions, and geometric relations. In the plastic analysis, rigid-plastic material behaviour based on simple J₂ flow theory is used.     

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.     

The effect of mechanical properties of sheet steels on the wrinkling behaviour during deep drawing of conical shells

Wrinkling is a common defect in press-formed parts, particularly in those with sloping walls. Its elimination often presents considerable technological problems. It is known that wrinkling can be prevented by stretching the material in the shell wall. This can be achieved by increasing the blankholder load but this is, of course, limited by the ductility of hte material in the cup wall. Only if the blankholder load is within a narrow range above that necessary to stretch out wrinkles on the one hand, and below that producing wall fracture on the other, will pressing be defect-free. The size of this workable range of blankholder loads determines the criticality of the pressing. The wider the range, the less critical the draw.It has been shown that for a given cup with sloping walls, the workable blankholder-load range cannot be significantly increased by modifying such forming conditions as blank size, die-profile radius and lubrication. A minor improvement can be achieved by increasing punch-profile radius, or by selected lubrication in the punch-nose area. In practice this means that, if the design or material thickness cannot be changed, the wrinkling problem can only be alleviated by using material with better mechanical properties.In the present investigation, the effect of mechanical properties on wall wrinkling in conical cups has been studied using cold-rolled and annealed stabilised steel, interstitial-free steel and high-strength low-alloy steel.The material with a higher n-value, higher r-value and a lower yield stress has generally been considered to be more wrinkle-resistant. The present investigation indicates that the effect of these properties is more complex. It has been found that the n-value influences the plastic-strain distribution during forming. In conical cups a high-n-value material develops higher compressive strains in the cup wall than a low-n-value material when pressed to the same depth, thus leading to earlier wrinkling. A high-r-value material has been found to be more wrinkle-resistant than a low-r-value material.Consequently, when selecting steel sheets for pressed parts where wrinkling is a problem, consideration has to be given to the effect of both r- and n-values. A material with high r_m-and low Δr-values should be specified. A high-n value material will not however alleviate the wrinkling problem in those cases where wrinkling is caused by excessive draw-in of material into the die orifice, such as in simple tapered cups, unless other changes in forming conditions, e.g., increased blankholder load, are also made. In complex parts, the effect of n-value on strain distribution should be evaluated with the grid-strain technique.     

Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells

Based on the nonlinear large deflection theory of cylindrical shells as well as the Donnell assumptions, this paper presents nonlinear buckling and postbuckling analyses for axially compressed functionally graded cylindrical shells by using the Ritz energy method and the nonlinear strain-displacement relations of large deformation. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Meanwhile, by taking into account the temperature-dependent material properties, various effects of external thermal environment are also investigated. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on nonlinear buckling and postbuckling behaviors. There is a mode-jumping observed after buckling. The present theoretical results are verified by those in the literature.     

Nonlinear acoustic diametral modes of pressure pulsations during flutter of rotor blades of a turbocompressor

Features of pressure pulsations of flow during flutter on the bending frequency of rotor blades of an axial turbocompressor, which occurs after incomplete resonance at the torsion frequency of blades with a harmonic of rotor speed, are investigated. The frequency spectrum of nonlinear acoustic diametral modes is predicted, based on Navier-Stokes equations ensemble-averaged. This makes it possible to identify characteristic series determined by the bending frequency of flutter and its second harmonic in the frequency spectrum for recording the pressure pulsation of flow during flutter along with harmonics of rotor speed. The occurrence the second harmonic of the bending frequency is the nonlinear reaction of flow, which locally limits the flutter amplitude in time.     

Nonlinear load-deflection relation of toroidal shells subjected to axisymmetric compression between rigid plates

Utility of toroidal shells as shock absorbers for radioactive materials' shipping casks is discussed by carrying out numerical and experimental analyses. The static load-deflection relations under axisymmetric compression between rigid plates are presented.A simple elastic-plastic analytical method is first developed on the basis of an incremental Rayleigh-Ritz method by taking account of the large deflections and the continuous change of contact points. Meridional compression is neglected and the meridional curvature distribution is approximated by a trigonometric series. Strain-hardening material properties and axisymmetric deformation are also assumed. In order to provide data for comparison purposes, static compression tests were performed using six models of different shapes and dimensions made of 304-type stainless steel. Finally, the applicability of the numerical method developed in this paper is discussed.The mechanical behavior of toroidal shells is also discussed.     

Nonlinear stability analysis of infinitely long laminated cylindrical shallow shells including shear deformation under lateral pressure

This paper presents a theoretical analysis for the various kinds of buckling behaviour of infinitely long laminated cylindrical shallow shells subjected to lateral uniform pressure. The exact solutions of the nonlinear equilibrium equations, in which first-order shear deformation is included, are obtained and the buckling criteria corresponding to different kinds of buckling are constructed taking into account the effects of the transverse shear deformation.     

圆锥滚子摆动从动件圆锥凸轮机构的廓面构建

以共轭曲面啮合理论为基础，运用矢量的旋转变换矩阵法，分析并推导了圆锥滚子摆动从动件圆锥凸轮机构的啮合方程和廓面方程，并给出了机构的压力角和综合曲率的计算公式，为圆锥凸轮加工精度的提高，改进其加工方法，实现凸轮CAD／CAM一体化奠定了坚实的理论基础。     

Identification of turbine blade flutter

Unstable vibrations during turbine flutter are studied. Analysis of strain gage records of blade vibrations and synchronous records of the pressure pulsations in the countercurrent gas flow shows that the damping of vibration modes is the most informative parameter determining flutter evolution. The time dependences of frequency spectra and of damping at these frequencies are determined by the Prony method. The coincidence of the eigenfrequencies of the blades and the synchronous character of the changes in the shape of the time dependences of damping at these frequencies and at the frequencies of the diametric modes of the pressure pulsations in the gas flow guarantee that it is collective vibrations of the blades related to the phenomenon of flutter that occur.