Localization of a delamination and estimation of its length in a composite laminate beam by the VSHM and pattern recognition methods

The purpose of this study was to investigate the delamination damage in laminate composite beams in order to adapt the vibration-based structural health monitoring (VSHM) method for laminated structures. The analysis was concentrated on the vibration characteristics of laminated specimens, in particular, on the first several natural frequencies of a composite laminate beam with a delamination damage. The core of this work is an experimental investigation into the vibration response of a composite laminate beam and its changes caused by delaminations of different sizes and different location in the beam. The aim was to determine how the first six harmonic frequencies are changed by a delamination, and the results show that they can be successfully used to clarify the presence, location, and dimensions of delaminations in a composite beam. A pattern recognition analysis was used to locate the damage, while its detection and evaluation were performed by using changes in the harmonic frequencies. A finite-element analysis was carried out, and the variations in the natural frequencies due to delamination are found to be in good agreement with experimental results.     

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

The vibrations of a thin elastic orthotropic circular cylindrical shell with free and hinged edges

The problem of the existence of natural oscillations of a thin elastic orthotropic circular closed cylindrical shell with free and hinge-mounted ends and of an open cylindrical shell with free and hinge-mounted edges, when the two boundary generatrices are hinge-mounted is investigated. Dispersion equations and asymptotic formulae for finding the natural frequencies of possible vibration modes are obtained using the system of equations corresponding to the classical theory of orthotropic cylindrical shells. A mechanism is proposed by means of which the vibrations can be separated into possible types. Approximate values of the dimensionless characteristic of the natural frequency and the attenuation characteristic of the corresponding vibration modes are obtained using the examples of closed and open orthotropic cylindrical shells of different lengths.     

Load and frequency interaction in dynamic buckling of soft-core sandwich plates and shells

The present study is concerned with an analysis of load-frequency interaction effects in the dynamic buckling response of soft-core sandwich plates and shells. Based on a higher-order geometrically non-linear sandwich shell theory in conjunction with an extended Galerkin procedure, a mathematical model together with analytical solution for simply supported sandwich plates and shells with with rectangular projection is obtained. By linearization of the non-linear solution with respect to the oscillating parts, the interaction of the natural frequencies with static preloads can be analyzed. The results reveal that the transverse compressibility of soft sandwich cores might have distinct effects on the natural frequencies. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution

This paper aims to investigate the free vibrational analysis of the generally doubly-curved shells of revolution made of functionally graded (FG) materials and constrained with different boundary conditions by means of an efficient, convenient and explicit method based on the Haar wavelet discretization approach. The FG materials of the shell consist of a combination of ceramic and metal, which four parameter power-law distribution functions have chosen for modeling of the smoothly and gradually variation of the material properties in the thickness direction. The theoretical model of the shell is formulated by employing of the first-order shear deformation theory. The rotation and displacement components of each point of the shell are expanded in the form of product of the Haar wavelet series in meridional direction as well as trigonometric series in the circumferential direction. By adding the boundary condition equations to the main system of equations, the constants appeared from the integrating of the Haar wavelet series are satisfied. In addition, with solving the characteristic equation, the vibrational results including the natural frequencies and the corresponding mode shapes are achieved. Then, the present results have been compared with those available in the literature. The results indicate that this method has high accuracy, high reliability and also a higher convergence rate in attaining the frequencies of the FG doubly-curved shells of revolution. Also, the effects of the main parameters such as power-law exponent, geometrical parameters, material distribution profiles and different types of boundary conditions, on the vibrational behavior of the FG doubly-curved shells of revolution, are investigated. Finally, taking into account the effects of geometrical parameters and material distribution profiles, for FG doubly-curved shells of revolution with different boundary conditions such as classic, elastic restraints and their combination, a variety of new frequency studies are provided which can be considered as proof results for further researches in this field.     

On Trapped Modes of Rotating Fluids in Spherical Shells

The question of existence of axisymmetric, equatorially trapped modes in rotating spherical shells (Stern, 1963; Stewartson and Rickard, 1969) is approached by means of a numerical simulation. The existence of one trapped mode is confirmed, and the dependence of its frequency on the thickness of the shell is investigated. The ray theoretical approach of Bretherton (1964) is also reconsidered, and it is found that in a shell of given thickness there are only a limited number of closed ray patterns which are confined to the vicinity of the equator. A continuous band of frequencies is associated with each one of these rays. It is found that the frequencies derived by the numerical simulation for thin shells agree with the maximum frequency in the first of these bands. It is conjectured that this fact may be associated with the viscous boundary conditions driving the forced oscillations inside the shell.     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

结构中的离散弹性支承

有几篇文章已经涉及结构的离散弹性支承.它与连续弹性支承有什么关系,两种支承情况对于结构动力特性产生多大差异,这是人们关心的问题.本文通过分析指出,频率是反应系统总能量中动能和势能的比例关系,离散支承和连续支承之间通过某种能量的等效转化,可保证频率不变.并给出了梁的理论推导和旋转壳的的数值结果.     

Free vibration analysis of two-dimensional functionally graded cylindrical shells

In this paper, the free vibration of a two-dimensional functionally graded circular cylindrical shell is analyzed. The equations of motion are based on the Love’s first approximation classical shell theory. The spatial derivatives of the equations of motion and boundary conditions are discretized by the methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ). Two kinds of micromechanics models, viz. Voigt and Mori–Tanaka models are used to describe the material properties. To validate the results, comparisons are made with the solutions for FG cylindrical shells available in the literature. The results of this study show that the natural frequency of the material can be modified in order to meet the expected results through manipulation of the constituent volume fractions. A comprehensive comparison is then drawn between ordinary and 2-D FG cylindrical shells.     

On the oscillations of a thin-walled structure containing a fluid, in the presence of a hydrodynamic damper : PMM vol. 43, no. 6, 1979, pp. 1095–1101

A model of a hydrodynamic oscillation damper is proposed. The model is used to obtain the equations describing longitudinal oscillations of a structure which includes a shell partially filled with fluid, and contains a hydrodynamic damper. It is shown that the use of the damper leads to considerable increase in the damping of the oscillations of specified frequencies within the structure.

In modern technology one encounters various types of problems connected with restricting the amplitudes of the axisymmetric vibrations of shells and of the longitudinal oscillations of structures consisting of shells partially filled with fluid. Various devices have been proposed [1] for solving these problems. All these devices have a common feature, namely an elastic shell filled with gas and placed in the fluid. The natural frequency of oscillations of such a shell in a fluid can be tuned to required frequency. The effect of such a device is analogous to the effect of a dynamic vibration damper in mechanical systems [2]. A part of the fluid contained in the shell serves as the active mass of the dynamic damper, and for this reason we shall call such devices the hydrodynamic vibration dampers.     

On one approach to studying free vibrations of cylindrical shells of variable thickness in the circumferential direction within a refined statement

We used the spline collocation method for finding the frequencies of free vibrations of circular closed cylindrical shells of variable thickness in the circumferential direction. The problem was formulated within the framework of Mindlin’s refined theory. We studied the influence of change in the shell thickness on the distribution of its natural frequencies. Our calculations were carried out for different geometrical parameters of the shell and different boundary conditions. The validity of results obtained was verified by increasing the number of collocation points in our calculations and by comparing them with the results of computations according to the three-dimensional theory.     

An experimeental study of the natural vibrations of supported shells

We give the results of an experimental study of the frequencies and shapes of the natural vibrations of shells of zero Gaussian curvature that are closed in the circumferential direction. The study was conducted using the method of holographic interferometry. We consider the influence of support by elastic ribs and attachment of rigid bodies on the dynamic properties of the shell system. We describe the methodology of the experiment. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 80–86.     

Calculation of the natural-vibration spectra of layered shells of moderate thickness

Conclusions Relations from a linear, kinematically nonuniform model of a layered shell were used to construct a system of motion equations for an M-layered shallow shell which considered all components of the stress-strain state and inertia of the shell. It was shown using sample calculations of the natural frequency spectrum of physically uniform and hybrid threelayer hells that this model makes it possible in a linear approximation to calculate the complete natural-frequency spectrum of layered shells. It can be used in engineering calculations of the dynamic characteristics of shells in which relatively thin and stiff bearing layers alternate in the packet with layers of a soft filler (structurally nonuniform hybrid shells).The use of simplified (classical) models, refined kinematically uniform models, and nonuniform models not accounting for compressive strains in the shell layers, etc. (see [1, 5]) is limited to the classes of physically uniform and quasiuniform shells and to cases of calculation of the dynamic characteristics determined by three fundamental frequencies of the shell when regarded as a three-dimensional body.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 298–304, March–April, 1985.     

Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow

New sandwich panels and tubes have widely applications in nanotechnology such as transportation, naval, aerospace industries, micro and nanoelectromechanical systems and fluid storage. For example, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In the current study, stability and free vibration analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow embedded in an orthotropic foundation under electro-thermo-mechanical loadings are presented. In fact, this work can be provided a valuable background for more research and further experimental investigation. It is assumed that the micro tubes are made of flexible material and smart piezoelectric composites reinforced by carbon nanotubes as core and face sheets, respectively. Energy method and Hamilton's principle are applied to derive the governing equations of motions based on Euler–Bernoulli beam model and using modified strain gradient theory. Moreover, generalized differential quadrature method is used to discretize and solve the governing equations of motions. Numerical results are investigated to predict the influences of length-to-radius, thickness of face sheets-to-thickness of core ratio, temperature changes, orthotropic elastic medium, Knudsen number, and carbon nanotubes volume fraction on the dimensionless natural frequencies and critical flow velocity of sandwich double-bonded piezoelectric micro composite tubes. The results of this article show that increasing the thickness ratio, volume fraction carbon nanotubes and orthotropic elastic constants lead to enhance the dimensionless natural frequency and stability of system, while decrease these parameters with increasing the temperature and length-to-radius ratio.     

Modelling of Wave-Based SHM Systems Using the Spectral Element Method

In nondestructive testing, the use of ultrasonic elastic waves has proven as one of the most successful techniques to detect structural damage such as cracks or delaminations. Especially, Structural Health Monitoring (SHM) is characterized by permanently installed actuators and sensors. As the capability of most approaches strongly depends on adequate choice of parameters like excitation signals and actuator/sensor positions, there is a growing interest in efficient and accurate simulation tools to be able to perform virtual SHM-system design. With respect to high frequency excitation an efficient simulation method is required. This contribution presents the formulation of spectral elements for flat shells based on the first order shear deformation theory (FSDT). A load-dependent geometrical stiffness matrix is included to take static preloads into account. The spectral element method generates an optimally concentrated mass matrix leading to significant savings of memory and to a reduction of complexity of the time integration algorithm. Dispersion curves of the FSDT are compared to those of 3D elasticity theory to determine the useful application range. Numerical and experimental examples for the propagation of waves including the influence of static prestress and the interaction with delamination are presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical-experimental investigation of frequencies of free vibrations of circular cylindrical shells

We consider a holographic interferometry technique for the determination of frequencies of free vibrations of isotropic circular cylindrical shells and compare the experimental results with results of calculation of frequencies of these shells on the basis of the spline collocation technique.     

Noncoaxial vibration of fluid-filled multi-walled carbon nanotubes

This paper is concerned with the free vibration of the fluid-filled multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on simplified Donnell’s cylindrical shell model and potential flow theory, the effect of internal fluid on the coupling vibration of the MWCNTs-fluid system is discussed in detail. The results show that the resonant frequencies are decreased due to the effect of the fluid, and the fluid has only a little influence on the associated amplitude ratio in MWCNTs corresponding to the natural resonant frequency (frequency of the innermost tube), while plays a significant role in the associated amplitude ratios corresponding to the intertube resonant frequency. For the natural resonant frequency, the vibration mode is coaxial. However, for the intertube resonant frequency, the system shows complex noncoaxial vibration, which plays a critical role in electronic and transport properties of carbon nanotubes (CNTs).     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.