Nonlinear vibrations of rotating thin circular cylindrical shell

Nonlinear vibrations of thin circular cylindrical shells are investigated in this paper. Based on Love thin shell theory, the governing partial differential equations of motion for the rotating circular cylindrical shell are formulated using Hamilton principle. Taking into account the clamped-free boundary conditions, the partial differential system is truncated by using the Galerkin method. Sequentially, the effects of temperature, geometric parameters, circumferential wave number, axial half wave number and rotating speed on the nature frequency of the rotating circular cylindrical shell are studied. The dynamic responses of the rotating circular cylindrical shell are also investigated in time domain and frequency domain. Then, the effects of nonlinearity, excitation and damping on frequency responses of steady solution are investigated.     

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10⁻⁹ and 0.95×10⁻⁹ m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e₀ for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.     

Free vibrations of rotating composite conical shells with stringer and ring stiffeners

In this paper, an analytical solution for the free vibration of rotating composite conical shells with axial stiffeners (stringers) and circumferential stiffener (rings), is presented using an energy-based approach. Ritz method is applied while stiffeners are treated as discrete elements. The conical shells are stiffened with uniform interval and it is assumed that the stiffeners have the same material and geometric properties. The study includes the effects of the coriolis and centrifugal accelerations, and the initial hoop tension. The results obtained include the relationship between frequency parameter and circumferential wave number as well as rotating speed at various angles. Influences of geometric properties on the frequency parameter are also discussed. In order to validate the present analysis, it is compared with other published works for a non-stiffened conical shell; other comparison is made in the special case where the angle of the stiffened conical shell goes to zero, i.e., stiffened cylindrical shell. Good agreement is observed and a new range of results is presented for rotating stiffened conical shells which can be used as a benchmark to approximate solutions.     

On dynamic buckling phenomena in axially loaded elastic-plastic cylindrical shells

Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range influence the initial instability pattern and the final buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t=0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy.     

Influence of fibre orientation and stacking sequence on petalling of glass/polyester composite cylindrical shells under axial compression

The energy absorbing capability of FRP composite cylindrical tubes used as energy absorbers, by destroying itself progressively, depends on the way in which the tube material is crushed i.e., trend of petalling. This paper investigates the influence of fibre orientation and stacking sequence on the petal formation and specific energy absorption (SEA) of four and six-ply, 0°/90° glass/polyester composite cylindrical shells under axial compression. Number of petals formed and the trend of petalling are changed with proportion of axial (0°) and circumferential (90°) fibre content and stacking sequence in the tubes. In the tubes undergo petalling, presence of axial fibres close to inner surface and the proper proportion of circumferential fibres close to outer surface of the tube wall lead to higher energy dissipation. The axial fibres placed nearer to outer surface leads to more number of petal formation, leading to a stable crushing mechanism. The contribution of mode I strain energy release rate (G_Ic) to the energy dissipation in the form of circumferential delamination is also studied with double cantilever beam (DCB) tests. Analytical model which considers petalling is developed, and used to predict the mean crush load and SEA of cylindrical composite shells under axial compression. Results from the analytical model agree well with experimental results and are presented.     

Effects of exponential volume fraction law on the natural frequencies of FGM cylindrical shells under various boundary conditions

In the present work, vibration characteristics of thin functionally graded cylindrical shells are studied under the influence of various boundary conditions. Fabrication of FGM cylindrical shell is carried out by using exponential volume fraction law. Strain- and curvature-displacements relationships are taken from Love’s thin shell theory. The frequency equation in the form of eigenvalue problem is obtained by adapting the Rayleigh-Ritz method. Characteristic beam functions are assumed to approximate the axial modal dependence. Effects of exponential volume fraction law on the natural frequencies of the FGM cylindrical shells for various boundary conditions are studied against circumferential wave number, length to radius ratio and thickness to radius ratio for different values of power law exponents. Results evaluated show good agreement with those available in the literature.     

含环向表面裂纹圆柱壳的波传播特性研究

研究了振动波在含有环向表面裂纹的无限长圆柱壳中的传播特性.圆柱壳体的振动用Flügge方程来描述.运用线弹性断裂力学的理论,考虑到裂纹的张开、滑移和撕裂3种模式以及它们相互之间的耦合,利用分布的线弹簧来模拟裂纹并建立了裂纹所在区域的局部柔度矩阵,得到由此引起的附加位移和壳体中内力之间的关系.在入射波已知的情况下,根据裂纹两侧区域的位移和内力的连续性条件得到了反射和透射波的幅值系数.分析了入射波通过裂纹后的透射、反射系数与激励频率和裂纹尺寸之间的关系.为基于振动功率流方法识别圆柱壳表面损伤提供了理论基础.     

爆炸处理消除大型柱壳结构环焊缝残余应力实验研究

为探讨大型耐压柱壳结构对接环焊缝焊接残余应力消除方法,根据耐压柱壳结构环焊缝焊接工艺,设计并制作了大型耐压柱壳结构对接环焊缝模拟焊接模型,并采用沿焊缝条状布药,分别进行了单、双面布药爆炸处理消除大型耐压柱壳结构对接环焊缝焊接残余应力模型实验。结果表明:大型耐压柱壳结构对接环焊缝存在较大的焊接残余应力;单、双面爆炸处理均能有效调整、消除焊接残余应力,残余应力分布明显均匀化;焊后残余应力越大,爆炸处理效果越明显,当焊接残余应力大于0.5s时,单面爆炸处理后残余应力m下降幅度在40%以上,双面爆炸处理后残余应力m下降幅度在60%以上;采用沿焊缝条状布药,爆炸处理能有效消除沿焊缝的纵向残余应力;双面爆炸处理效果优于单面爆炸处理效果。     

含裂纹损伤充液圆柱壳的振动响应求解方法

薄壁圆柱壳流体冲击振动响应是一个复杂的流固耦合(FSI)动力学问题,对于薄壳状态监测与缺陷识别具有重要意义。基于Flügge壳体应力理论,得到壳体运动的高阶偏微分方程组(PDE),利用波传播方法获得圆柱壳系统振动响应。将壳体周围流体定义为理想声学介质,通过亥姆霍兹方程描述声压场,得到流固耦合条件下的薄壁圆柱壳受迫振动响应演变规律。针对薄壳裂纹损伤识别问题,基于断裂力学理论建立局部柔度矩阵,结合呼吸型线弹簧模型(LSM),构造裂纹附近应力及位移连续条件,获得含裂纹损伤充液圆柱壳的振动响应,进而给出一种基于振动能量流的裂纹损伤识别方法。研究结果表明：充液圆柱壳耦合系统在非线性激励下,位移响应在沿轴向、周向和径向的传播特性差异明显;裂纹的存在会导致结构局部柔度的降低和耦合系统固有频率下降;归一化输入功率流能够有效地对充液圆柱壳耦合系统进行结构裂纹损伤识别。研究结果可为充液薄壳振动响应方面的研究提供有益参考,也可为流固耦合条件下的结构裂纹损伤识别方面提供技术支持。     

Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid

Finite amplitude combined axial and torsional shear wave propagation in an incompressible isotropic hyperelastic solid is considered. When the strain energy function of the solid is a non-linear function offI₁,− 3) and (I₂− 3), where I₁, and I₂are the first and second basic invariants of the left Cauchy-Green tensor, the two second order partial differential equations governing the propagation of the axial and torsional waves are non-linear and coupled. These two coupled equations are equivalent to a hyperbolic system of first order partial differential equations and a modification of the MacCormack finite difference scheme is used to obtain numerical solutions of this system. Numerical results, which show the effect of the coupling, are presented for boundary-initial value problems of propagation into initially unstressed and initially stressed regions at rest.     

Three‐dimensional numerical simulation of Marangoni instabilities in liquid bridges: influence of geometrical aspect ratio

Oscillatory Marangoni convection in silicone oil–liquid bridges with different geometrical aspect ratios is investigated by three‐dimensional and time‐dependent numerical simulations, based on control volume methods in staggered cylindrical non‐uniform grids. The three‐dimensional oscillatory flow regimes are studied and compared with previous experimental and theoretical results. The results show that the critical wavenumber (m), related to the azimuthal spatio‐temporal flow structure, is a monotonically decreasing function of the geometrical aspect ratio of the liquid bridge (defined as the ratio of length to diameter). For this function, a general correlation formula is found, which is in agreement with the previous experimental findings. The critical Marangoni number and the oscillation frequency are decreasing functions of the aspect ratio; however, the critical Marangoni number, based on the axial length of the bridge, does not change much with the aspect ratio. For each aspect ratio investigated, the onset of the instability from the axisymmetric steady state to the three‐dimensional oscillatory one is characterized by the appearance of a standing wave regime that exhibits, after a certain time, a second transition to a travelling wave regime. The standing wave regime is more stable for lower aspect ratios since it lasts for a long time. This behaviour is explained on the basis of the propagation velocity of the disturbances in the liquid phase. For this velocity, a general correlation law is found as a function of the aspect ratio and of the Marangoni number. Copyright © 2001 John Wiley & Sons, Ltd.     

A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field

With the aid of a generalized variational method, in this paper, a theoretical model for soft ferromagnetic shells is derived to describe their magnetoelastic behavior in an applied magnetic field. Having made a quantitative comparison between the numerical predictions given by several theoretical models and the experimental results on strains of a cylindrical shell, we find that the predictions got by our model are in good agreement with the experimental data. It is also found that the Moon’s model is a special case of the model derived in this paper when the relative magnetic permeability μ_r>10⁴, which confirms that it is reasonable for the Moon’s model to calculate strains of the soft ferromagnetic shells. Having displayed the distribution of the equivalent magnetic force in the length of the cylindrical shell and its circumferential bending strains with different elastic end constraints, we give an explanation for the discrepancy between Moon’s analytical results and his experimental ones.     

Reconstruction of the three mechanical material constants of a lossy fluid-like cylinder from low-frequency scattered acoustic fieldsReconstruction des trois constants mécaniques matériels d'un cylindre fluide dissipatif à partir de champs acoustiques basses fréquences

The inverse medium problem for a circular cylindrical domain is studied using low-frequency acoustic waves as the probe radiation. To second order in k₀a (k₀ the wavenumber in the host medium, a the radius of the cylinder), only the first three terms (i.e., of orders 0, ?1 and +1) in the partial wave representation of the scattered field are non-vanishing. This enables the scattered field to be expressed algebraically in terms of the unknown material constants, i.e., the density ρ₁, and the real and imaginary parts of complex compressibility κ₁ of the cylinder. It is shown that these relations can be inverted to yield explicit, decoupled expressions for ρ₁ and κ₁ in terms of the totality of the far-zone scattered field. These expressions furnish accurate estimations of the material parameters provided the probe frequency is low and the radius of the cylinder is known very precisely. To cite this article: T. Scotti, A. Wirgin, C. R. Mecanique 332 (2004).     

Natural frequencies of rotating functionally graded cylindrical shells

Love’s first approximation theory is used to analyze the natural frequencies of rotating functionally graded cylindrical shells.To verify the validity of the present method,the natural frequencies of the simply supported non-rotating isotropic cylindrical shell and the functionally graded cylindrical shell are compared with available published results.Good agreement is obtained.The effects of the power law index,the wave numbers along the x-and θ-directions,and the thickness-to-radius ratio on the natural frequencies of the simply supported rotating functionally graded cylindrical shell are investigated by several numerical examples.It is found that the fundamental frequencies of the backward waves increase with the increasing rotating speed,the fundamental frequencies of the forward waves decrease with the increasing rotating speed,and the forward and backward waves frequencies increase with the increasing thickness-to-radius ratio.     

加权残值法分析轴压圆柱薄壳后屈曲问题

本文首次应用了样条配点法分析了受到轴向压力的圆柱形薄壳的后屈问题,壳体的方程是L.H.Donnell的非线性正交异性圆柱形壳体方程,壳体的挠度试函数及应力函数试函数都是于轴向用了五次B样条函数基函数,周向用余弦函数。计算模型是周向为半个波长的壳块,可适应后屈曲实验变形跳跃现象。非线性代数方程组用了Newton—Rophson迭代法求解。由此所得的理论上的后屈曲曲线与国外近代实验相符。     

VIBRATIONS OF CIRCULAR CYLINDRICAL SHELLS WITH NONUNIFORM CONSTRAINTS,ELASTIC BED AND ADDED MASS; PART I: EMPTY AND FLUID-FILLED SHELLS

Circular cylindrical shells with non uniform edge constraints (with zero radial and circumferential displacement) are investigated, including riveted shells. The linear modes of simply supported shells vibrating in vacuo are used as admissible functions, and the solution is obtained with the artificial spring method. The Flügge theory of shells is used and in-plane inertia is retained. Any shell constraint other than simple supports can be studied with the proposed method. Complicating effects due to the contained inviscid fluid, elastic bed of partial axial and angular dimensions, intermediate constraints and added mass are considered. The convergence of the method is numerically investigated and the effect of the number of rivets (clamped arcs) on shell modes is studied.     

Transient radiation by a submerged fluid-filled cylindrical shell

The radiation by a submerged fluid-filled cylindrical shell in response to a transient external pressure pulse is considered, and a semi-analytical model based on the Reissner–Mindlin shell theory is employed to simulate the interaction numerically. Two types of radiated waves that have been previously seen in experimental images for a submerged evacuated cylindrical shell are observed in both the external and internal fluids, the symmetric Lamb waves S₀ and the antisymmetric Lamb (or pseudo-Rayleigh) waves A₀. The third type of radiated waves is also observed that has not been explicitly imaged either experimentally or numerically for a submerged evacuated cylindrical shell, and it is demonstrated that these waves are the Scholte–Stoneley waves A. The effect that the complex structure of the radiated field has on the wave phenomena in the internal fluid is analyzed for shells of several different thicknesses, and the results of this analysis are summarized in the form of diagrams suitable for the use at the pre-design stage.     

Wave transmission through laminated composite double-walled cylindrical shell lined with porous materials

A study on free harmonic wave propagation in a double-walled cylindrical shell, whose walls sandwich a layer of porous materials, is presented within the framework of the classic theory for laminated composite shells. One of the most effective components of the wave propagation through the porous core is estimated with the aid of a fiat panel with the same geometrical properties. By considering the effective wave component, the porous layer is modeled as a fluid with equivalent properties. Thus, the model is simplified as a double-walled cylindrical shell trapping the fluid media. Finally, the transmission loss (TL) of the structure is estimated in a broadband frequency, and then the results are compared.     

Buckling in Space and Time

A personal overview of the buckling process is given, based on more than 30 years research experience. A fundamental distinction is drawn between structures that tend to distribute their pattern of buckling over the entire available length, and those that favour limiting it to a localized portion. Difficulties are compounded by the fact that localized solutions typically are shadowed by distributed counterparts. The difference is found to depend primarily on the stability of the system at the point of buckling. When an unstable localized form shows a tendency to restabilize, a pattern known as cellular buckling can develop, taking place sequentially in both space and time. The paper focuses particularly on the buckling of the axially-compressed cylindrical shell, and shows how the circumferential periodicity coupled with axial localization has special but hidden characteristics. In particular, the circumferential wavenumber is picked at an early stage of buckling and remains locked, while localization in the axial sense permits extensive change to the post-buckling shape.     

Instability of small-amplitude convective flows in a rotating layer with free boundaries

The stability of steady convective flows in a horizontal layer with free boundaries, heated from below and rotating about a vertical axis, is studied in the Boussinesq approximation (Rayleigh-Bénard convection). The flows considered are convective rolls or square cells that are sums of two perpendicular rolls with equal wave numbers k. It is assumed that the Rayleigh number is almost critical in order for convective flows with a wave number k: R = R _c (k) + ε² to arise, the amplitude of the supercritical states being of the order of ε. It is shown that the flows are always unstable relative to perturbations that are the sum of one long-and two short-wave modes corresponding to linear rolls turned through small angles in opposite directions.