Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.     

Flapwise bending vibration analysis of rotating composite cantilever beams

A modeling method for the modal analysis of a rotating composite cantilever beam is presented in this paper. Linear differential equations of motion are derived using the assumed mode method. For the modeling, hybrid deformation variables are employed and approximated to derive the equations of motion. Symmetrical laminated composite beams are considered to obtain the numerical results. The effects of the dimensionless angular velocity, the hub radius and the fiber orientation angle on the variations of modal characteristics are investigated.     

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.     

A high-order shear element for nonlinear vibration analysis of composite layered plates and shells

This paper introduces a family of high-order facetted shell elements for linear and nonlinear stress and vibration analysis of composite layered plate and shell structures. Engineering slope angles are employed in element equations, and transverse stresses are expanded over the thickness. The lateral deflection is modelled by conforming or non-conforming Hermitian shape functions, within rectangular or paralellogrammic and triangular elements. Nonlinear terms associated with geometrical nonlinearity are also derived using a practical approach based upon the actual components of strain. A finite element programming package was designed employing the newly developed elements. Several case studies have been investigated and package results were compared with existing theoretical and/or experimental results. It has been proved that the developed elements can lead to accurate estimations of natural frequencies. The effect of fibre angles on natural frequencies has also been investigated with some case studies, and the results proved that the package can be a useful tool for the design optimization of composite layered plate and shell structures.     

An analysis of the free vibration of thick-walled isotropic toroidal shells

The equations of motion and strain–displacement equations for toroidal shells are derived using toroidal coordinates and dyadic methods of elasticity. Suitable assumptions are proposed that reduce the governing equations to axisymmetric format while maintaining the three-dimensional character of the solution. The free vibration analysis of toroidal shells is carried out using numerical solutions of the governing equations. A nine-node Lagrangian finite element is formulated in the toroidal coordinate system and solutions are obtained for the case where an axis of symmetry can be assumed at the center of the torus. Results for frequency of vibration are tabulated for solid and thick-walled shells. Mode shapes are shown for a representative thick-walled torus.     

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.     

A Vectorial-Wave Method for free and forced vibration analysis of extra thin cylindrical shells with boundary discrete damping

The Vectorial-wave method (VWM) is developed to study free and forced vibrations of cylindrical shells in the presence of dampers at supports. In modeling the issue, a circular cylindrical shell is considered with two ended supports, including separate springs and viscous dampers in the possible directions. Accordingly, based on Flügge thin shell theory and by considering the wave vectors going in the opposite direction along with the shell axis, reflection and transmission matrices are determined to satisfy the shell continuity as well as the boundary conditions. The proposed method is verified through comparing its results with the available literature and the numerical results calculated by Finite element method (FEM). Employing VWM, the viscous characteristics of the applied supports on natural frequencies of the shell are investigated. Furthermore, frequency responses of the shell, which are affected by point-load excitation, are obtained. Finally, the results show that several tandem resonance picks can be eliminated via accurate setting of the support damping.     

A study on free vibration of a spinning disk

Without a logical jump, we have derived the governing equation for free vibration of a spinning circular disk by using the variational formulation based upon the Kirchhoff plate theory and von Karman strain one. It has been found during the derivation that the governing equation is theoretically valld under the assumption that in plane deflections are steady and axisymmetric, and that internal forces are linearized while the strains remain nonlinear. The natural frequencies and the critical speeds of a freely spinning disk are obtained approximately and their dependencies on the spinning speed, mode number, and natural freqeuncy of the stationary disk are analyzed.     

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.     

Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions

This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.     

Thermoelastic and vibration analysis of functionally graded cylindrical shells

The static response and free vibration of metal and ceramic functionally graded shells are analyzed using the element-free kp-Ritz method. The material properties are assumed to vary continuously along the depth direction. The displacement field is expressed in terms of a set of mesh-free kernel particle functions according to Sander's first-order shear deformation shell theory. The effects of the volume fraction, material property, boundary condition, and length-to-thickness ratio on the shell deflection, axial stress, and natural frequency are examined in detail. Convergence studies of node numbers are performed to verify the effectiveness of the proposed method. Comparisons reveal that the numerical results obtained from the proposed method agree well with those from the classical and finite element methods.     

Vibration analysis of delaminated composite beams and plates using a higher-order finite element

In order to analyze the vibration response of delaminated composite plates of moderate thickness, a FEM model based on a simple higher-order plate theory, which can satisfy the zero transverse shear strain condition on the top and bottom surfaces of plates, has been proposed in this paper. To set up a C⁰-type FEM model, two artificial variables have been introduced in the displacement field to avoid the higher-order derivatives in the higher-order plate theory. The corresponding constraint conditions from the two artificial variables have been enforced effectively through the penalty function method using the reduced integration scheme within the element area. Furthermore, the implementation of displacement continuity conditions at the delamination front has been described using the present FEM theory. Various examples studied in many previous researches have been employed to verify the justification, accuracy and efficiency of the present FEM model. The influences of delamination on the vibration characteristic of composite laminates have been investigated. Especially the variation of ‘curvature of vibration mode’ (i.e., the second-order differential of deflections in vibration mode) caused by delamination has been studied in detail to provide valuable information for the possible identification of delamination. Furthermore, two approaches have been investigated to detect a delamination in laminates by employing this information.     

A study on stress analysis of orthotropic composite cylindrical shells with a circular or an elliptical cutout

The stress analysis on orthotropic composite cylindrical shells with one circular or one elliptical cutout subjected to an axial force is carried out by using an analytical and experimental method. The composite cylindrical shell governing equation of the Donnell's type is applied to this study and all results are presented by the stress concentration factor. The stress concentra-tion factor is defined as the ratio of the stress on the region around a cutout to the nominal stress of the shell. The stress concentration factor is classified into the circumferential stress concen-tration factors and the radial stress concentration factors due to the cylindrical coordinate of which the origin is the center of a cutout. The considered loading condition is only axial tension loading condition. In this study, thus, the maximum stress is induced on perpendicular region against axial direction, on the coordinate. Various cutout sizes are expressed using the radius ratio,, which is the radius of a cutout over one of the cylindrical shell. Experimental results are obtained using strain gages, which are attached around a cutout of the cylindrical shell. As the result from this study, the stress concentration around a cutout can be predicted by using the analytical method for an orthotropic composite cylindrical shell having a circular or an elliptical cutout.     

Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method

In the present study, a spline finite strip with higher-order shear deformation is formulated for stability and free vibration analysis of piezoelectric composite plates. At each knot, the electric potentials on the surfaces and middle plane of each piezoelectric layer are taken as nodal degrees of freedom. However, if a continuous electrode is installed on the surface of the layer, the electric potential on the electrode is changed to structural degree of freedom, so that the equipotential condition on the electrode is automatically satisfied. The analysis can be conducted based on Reddy's third-order shear deformation theory, Touratier's “Sine” model, Afaq's exponential model or Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick plates over the first-order shear deformation theory is achieved at only little extra computational cost.The numerical results obtained based on different shear deformation theories are presented in comparison with the three-dimensional solutions. The effects of length-to-thickness ratio, fiber orientation, boundary conditions and electrical conditions on the natural frequency and critical buckling load of piezoelectric composite plates are investigated through numerical examples.     

面向叶片弯扭变形分析的测量采样方法

针对叶片弯扭变形分析的问题,对等参数、等弧长及等弦高测量采样方法进行了研究.采用以上三种测量采样方法对叶片CAD截面线进行离散,对得到的离散点集进行平移、旋转及加噪处理,作为叶片弯扭变形的截面测量数据;通过采用奇异值分解和最近点迭代相结合的算法,将叶片截面测量数据与计算机辅助设计模型的截面轮廓线进行配准,计算出测量截面线相对于CAD模型截面线的扭转角度和平移量,并得出不同测量采样方法对应的叶片弯扭变形分析的精度.通过基于仿真数据和实测数据的叶片截面模型配准实验及对比分析,总结出一些可用于指导叶片弯扭变形分析的测量采样策略.     

Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells

Based on the three-dimensional fundamental equations of anisotropic elasticity, a state equation with variable coefficients is derived in a unified matrix form. The free vibration of simply supported, fluid-filled cylindrically orthotropic functionally graded cylindrical shells with arbitrary thickness is then investigated. A laminate approximate model is employed which is suitable for an arbitrary variation of material constants along the radial direction. Numerical examples are presented and compared with existing results. The effects of related parameters on natural frequencies are discussed finally.     

A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams

This paper uses He’s Homotopy Perturbation Method (HPM) to analyze the nonlinear free vibrational behavior of clamped-clamped and clamped-free microbeams considering the effects of rotary inertia and shear deformation. Galerkin’s projection method is used to reduce the governing nonlinear partial differential equation. to a nonlinear ordinary differential equation. HPM is used to find analytic expressions for nonlinear natural frequencies of the pre-stretched microbeam. A parametric study investigated the effects of design parameters such as applied axial loads and slenderness ratio. The effect of rotary inertia and shear deformation on the nonlinear natural frequency was investigated. For verification, a numerical approach was implemented to solve the nonlinear equation. of vibration. A comparison between analytical and numerical results shows that HPM can predict system nonlinear vibrational behavior significantly more accurately than previously used methods in the literature.     

A comparative experimental study on the use of acoustic emission and vibration analysis for bearing defect identification and estimation of defect size

Vibration monitoring of rolling element bearings is probably the most established diagnostic technique for rotating machinery. The application of acoustic emission (AE) for bearing diagnosis is gaining ground as a complementary diagnostic tool, however, limitations in the successful application of the AE technique have been partly due to the difficulty in processing, interpreting and classifying the acquired data. Furthermore, the extent of bearing damage has eluded the diagnostician. The experimental investigation reported in this paper was centred on the application of the AE technique for identifying the presence and size of a defect on a radially loaded bearing. An experimental test rig was designed such that defects of varying sizes could be seeded onto the outer race of a test bearing. Comparisons between AE and vibration analysis over a range of speed and load conditions are presented. In addition, the primary source of AE activity from seeded defects is investigated. It is concluded that AE offers earlier fault detection and improved identification capabilities than vibration analysis. Furthermore, the AE technique also provided an indication of the defect size, allowing the user to monitor the rate of degradation on the bearing; unachievable with vibration analysis.     

Nonlinear free vibration of laminated composite plates on elastic foundation with random system properties

The present investigation is concerned with free vibration analysis of laminated composite plates resting on elastic foundation undergoing large amplitude oscillation with random system properties. The lamina material properties and foundation stiffness parameters are modeled as basic random variables for accurate prediction of the system behavior. The basic formulation of the problem is based on higher-order shear displacement theory including rotatory inertia effects and von Karman-type nonlinear strain displacement relations. A C⁰ finite element is used for descretization of the laminate. A direct iterative method in conjunction with first-order Taylor series based perturbation technique procedure is developed to solve random nonlinear generalized eigenvalue problem. The developed probabilistic procedure is successfully used for the nonlinear free vibration problem with a reasonable accuracy. Typical numerical results (second-order statistics) are obtained for the composite plates resting on Winkler and Pasternak elastic foundations with different support conditions, side-to-thickness ratio, aspect ratio, oscillation amplitude ratio, stacking sequences and foundation parameters for symmetric and anti-symmetric cross-ply and angle-ply laminates. The results are validated with existing available results and independent Monte Carlo simulation.     

A comparative experimental study on the diagnostic and prognostic capabilities of acoustics emission,vibration and spectrometric oil analysis for spur gears

Prognosis of gear life using the acoustic emission (AE) technique is relatively new in condition monitoring of rotating machinery. This paper describes an experimental investigation on spur gears in which natural pitting was allowed to occur. Throughout the test period, AE, vibration and spectrometric oil samples were monitored continuously in order to correlate and compare these techniques to natural life degradation of the gears. It was observed that based on the analysis of root mean square (rms) levels only the AE technique was more sensitive in detecting and monitoring pitting than either the vibration or spectrometric oil analysis (SOA) techniques. It is concluded that as AE exhibited a direct relationship with pitting progression, it offers the opportunity for prognosis.