Free vibration of elastic helicoidal shells

An elastic helicoidal structure modelled as a plate twisted around its axis is studied in this paper. Accurate strain–displacement relationships for the shell are derived by the Green strain tensor in general shell theory and first-order shear deformation theory. An energy equilibrium equation of free vibration is introduced by the principle of virtual work. Applying the Rayleigh–Ritz method, an analytical eigenvalue equation is formulated and solved via an efficient computational approach for vibration characteristics of the helicoidal structure. A set of normalized orthogonal polynomials generated by the Gram–Schmidt procedure is presented to approximate the admissible functions. The first polynomial is taken as a kinematically compliant geometric equation of boundary conditions of the shell. The convergence and the accuracy of the present method, and the effects of geometric parameters and boundary conditions on vibration of the helicoidal structure are investigated.     

双材平面中环形界面环向裂纹问题的超奇异积分方程方法

从研究环形界面双相材料平面任点处沿径向、环向作用单位力时的弹性力学基本解出发,利用Betti定律、几何关系和虎克定律得到双材料平面环向裂纹问题的位移场和应力场表达式,经代入裂纹岸应力边界条件,导出极坐标下以裂纹岸位移间断为基本未知量的超奇异积分方程组;通过适当的积分变换,用有限部积分原理处理方程组中所包含的两类奇异积分—Cauchy奇异积分和超奇异积分,解决极坐标下环形界面双材料平面环向裂纹问题用超奇异积分方程法的理论描述与数值算法。在嵌入物半径足够大时,计算结果与已发表文献对直线界面情况下平行于界面裂纹问题的计算结果一致。     

Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

Frequency analysis of multiple layered cylindrical shells under lateral pressure with asymmetric boundary conditions

Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love＇s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped（SS-C）, free-clamped（F-C） and simply supported-free（SS-F）. The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.     

Using Eigenvalues of Covariance Matrices for Automated Visual Inspection of Microdrills

This paper proposes a translation, rotation, and template-free automated visual inspection scheme that detects microdrill defects using the eigenvalues of covariance matrices. We first derived the colour images of microdrills and extracted the boundary of the first facets. Then, the smaller eigenvalues of the covariance matrices of given regions of support were calculated for boundary representation, and they were thresholded to separate the boundaries into segments. The least square linear regression method was used to fit the segments into linear equations. Eventually, the defects were detected by three inspection rules that measure five features of microdrills including: gap distance, parallel, and enclosed angles, accordingly. The proposed scheme was implemented in C++ with a graphical user interface environment. Fifteen microdrills, digitized without alignment, were used to verify the proposed inspection process. Experimental results show that the proposed scheme reliably achieves the inspection of microdrills.     

Shape design sensitivity analysis for interface problem in axisymmetric elasticity

A boundary integral equation method in the shape design sensitivity analysis is developed for the elasticity problems with axisymmetric non-homogeneous bodies. Functionals involving displacements and tractions at the zonal interface are considered. Sensitivity formula in terms of the interface shape variation is then derived by taking derivative of the boundary integral identity. Adjoint problem is defined such that displacement and traction discontinuity is imposed at the interface. Analytic example for a compound cylinder is taken to show the validity of the derived sensitivity formula. In the numerical implementation, solutions at the interface for the primal and adjoint system are used for the sensitivity. While the BEM is a natural tool for the solution, more generalization should be made since it should handle the jump conditions at the interface. Accuracy of the sensitivity is evaluated numerically by the same compound cylinder problem. The endosseous implant-bone interface problem is considered next as a practical application, in which the stress value is of great importance for successful osseointegration at the interface. As a preliminary step, a simple model with tapered cylinder is considered in this paper. Numerical accuracy is shown to be excellent which promises that the method can be used as an efficient and reliable tool in the optimization procedure for the implant design. Though only the axisymmetric problem is considered here, the method can be applied to general elasticity problems having interface.     

Vibration of cylindrical shells with ring support

In this paper, a study on the vibration of thin cylindrical shells with ring supports is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using Sanders' shell theory. The governing equations are obtained using an energy functional with the Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.     

基于高温铸坯黏弹塑性的坯壳动态鼓肚研究

基于高温铸坯材料黏弹塑性本构方程,建立了坯壳动态鼓肚数学模型。根据坯壳不同的初始状态与变形历史确定了两种边界条件。利用模型计算坯壳鼓肚变形,并与实测数据进行对比,验证了理论解的正确性以及模型的有效性。根据鞍钢工业板坯连铸机的设备工艺参数,分别计算了刚出结晶器的坯壳和远离结晶器的坯壳的鼓肚变形曲线,以及坯壳在固液交界面处的应变与应变速率。分析了铸坯坯壳在铸机扇形段内的鼓肚变形与应变变化规律,并讨论了辊间距、     

Free vibration analysis of joined conical-cylindrical shells by matched Fourier-Chebyshev collocation method

A free vibration analysis of joined conical-cylindrical shells based on the first order shear deformation theory is developed. Deflections and rotations are represented by the expansions of Chebyshev polynomials and Fourier series. Equations of motion are collocated to yield the system of algebraic equations. Boundary conditions and compatibility conditions are considered as side constraints, and the set of algebraic equations is condensed so that the number of degrees of freedom matches the number of expansion coefficients. Numerical examples are provided for a joined conical-cylindrical shell, a complete conical shell attached to a cylindrical shell and a hermetic can.     

Simulation of sloshing in a bi-lobe tank under arbitrary rotation using the FDS scheme and the HCIB method

Three-dimensional sloshing in a bi-lobe tank under arbitrary rotation is simulated using a code developed using the flux-difference splitting scheme for variable density incompressible fluids and the hybrid Cartesian/immersed boundary method. The material interface is regarded as a moving contact discontinuity and is captured using a free surface capturing method derived from the Riemann solver, without any additional treatment along the interface. The boundary condition for the arbitrary motion of the bi-lobe tank, which contains a thin partition between two partially overlapping cylindrical tanks, is handled with ease by using the hybrid Cartesian/immersed boundary method. The computed time evolution of the interface is compared with the snapshots taken during the experiments on sloshing caused by the sway motion of the bi-lobe tank. Good agreement is observed between the computational and experimental results. The validated code is used to simulate three-dimensional sloshing in the bi-lobe tank that is subject to combined pitch and roll motions. A rotational vector is used to locate the Lagrangian points of the unstructured surface grid according to the motion of the tank. Grid independence tests are carried out using three different size grids. Violent three-dimensional sloshing ensues with an increase in the angular velocity of rotation.     

Frequency response analysis of cylindrical shells conveying fluid using finite element method

A finite element vibration analysis of thin-watled cylindrical shells conveying fluid with uniform velocity is presented The dynamic behavior of thin-walled shell is based on the Sanders’ theory and the fluid in cyhndrical shell is considered as inviscid and incompressible so that it satisfies the Laplace’s equation A beam-like shell element is used to reduce the number of degrees-of-freedom by restricting to the circumferential modes of cylindrical shell An estimation of frequency response function of the pipe considering of the coupled effects of the internal fluid is presented A dynamic coupling condition of the interface between the fluid and the structure is used The effective thickness of fluid according to circumferential modes is also discussed The influence of fluid velocity on the frequency response function is illustrated and discussed The results by this method are compared with published lesults and those by commercial tools     

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.     

利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。     

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.     

Natural oscillations of general shells based on nonclassical theory

Two-dimensional equations of the dynamics of the theory of general shells and appropriate boundary conditions that make it possible to take into account the transverse shear and compression of the shell are constructed based on three-dimensional equations of elasticity theory and the Lagrange variational principle by expanding the displacements in the coordinate normal to the middle surface. Natural oscillations of a circular cylindrical shell are considered. Frequencies of natural oscillations are determined by the Bubnov-Galerkin method. The impact of different types of boundary conditions and geometric parameters of the shell on the value of natural frequencies are analyzed. Simulation results are compared with different variants of the classical theory of shells, as well as with three-dimensional elasticity theory.     

Free vibration analysis of a hermetic capsule by pseudospectral method

A pseudospectral method is applied to the axisymmetric free vibration analysis of a hermetic capsule. The displacements and the rotation of a hermetic capsule are expressed by the Chebyshev expansions. The equations of motion are collocated to yield the system of equations in the hemispherical regions and the cylindrical region separately. The numbers of collocation points are chosen to be less than those of the expansion terms. The continuity conditions of deformations and stress resultants at the junctions serve as the constraints of the expansion coefficients. The set of algebraic equations is condensed so that the total number of the expansion terms matches the total number of degrees of freedom of the problem. Present method might be useful in the analyses of composite shell structures where different types of shells are joined together.     

Analysis of the vibration behavior of FGM cylindrical shells including internal pressure and ring support effects based on Love-Kirchhoff theory with various boundary conditions

This paper presents the study on vibration behavior of functionally graded material (FGM) cylindrical shell with the effects of internal pressure and ring support. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with internal pressure and ring support are established based on strain-displacement relationship using Love-Kirchhoff shell theory. The governing equations of motion were solved by using energy functional and by applying Ritz method. The boundary conditions represented by end conditions of the FGM cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), free-simply supported (F-SS), free-sliding (F-SL) and clamped-sliding (C-SL). To check the validity and accuracy of the present method, the results obtained are compared with those available in the literature. The influence of internal pressure, ring support position and the effect of the different boundary conditions on natural frequencies characteristics are studied. These results presented can be used as important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with internal pressure and ring support.     

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.     

Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions

In the present work, study of the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. Effects of boundary conditions and ring support on the natural frequencies of the FGM cylindrical shell are studied. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using different shear deformation shell theories. The analysis is carried out using Hamilton’s principle. The governing equations of motion of a FGM cylindrical shells are derived based on various shear deformation theories. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin M. M. Najafizadeh received his BS degree in 1995 from Azad University (Arak) and the Ms Degree in 1997 from Azad University (Arak), and his Ph.D. degree in 2003 from Science and Research Branch Islamic Azad University (Tehran, Iran), all in mechanical Engineering. He is member of faculty in Islamic Azad University (Arak) since 1998. He teaches courses in the areas of dynamics, theory of plates and shells and finite element method. He has published more than 20 articles in journals and conference proceeding. Mohammad Reza Isvandzibaei received his Ms Degree from Azad University (Arak), and now he is the student of Ph.D. in university of Pune, (India) all in mechanical Engineering. He is member of faculty in Islamic Azad University (Andimeshk).     

柔性臂动力学模型的比较研究

柔性臂系统控制经常采用的是简支梁模型和悬梁模型 ,但对哪一种模型更准确的问题尚无统一的认识。本文建立了单柔性臂系统的间支梁模型、悬臂梁模型 ,求出了它们在几种条件下的广义应变响应 ,并与有限元模型的结果进行比较。仿真结果表明 ,悬臂梁模型与有限元模型具有较高的一致性 ,它比简支梁模型更准确