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.     

Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions

Based on the Reissner-Naghdi’s thin shell theory, this paper concentrates on the free vibration of a joined cylindrical-spherical shell with elastic-support boundary conditions by a domain decomposition method. The joined shell is first separated from the elastic-support boundary and then subdivided into some cylindrical and spherical shell segments along the axis of revolution. The elastic-support boundary is regarded as a combination of distributed linear springs and can be treated as a special interface as well as the interface between two adjacent shell segments. Through the variational operation of the whole energy functional, the discretized equations of motion are derived by the expansions of the displacement field for each shell segment with Fourier series and Chebyshev orthogonal polynomials in the circumferential and longitudinal direction, respectively. To analyze the numerical convergence and precision of the present method, a number of case studies have been conducted and the solutions are compared with those derived by ANSYS and those presented in the previous literature to confirm the reliability and accuracy.     

The application of the pseudospectral method to the analysis of helical springs of arbitrary shape

The pseudospectral method is applied to the static analysis of helical springs of arbitrary shape. The displacements and the rotations are approximated by series expansions of Chebyshev polynomials. The entire domain is considered as a single element and the governing equations are collocated to yield the system of algebraic equations. The boundary conditions are considered as the constraints, and the set of equations is condensed so that the number of degrees of freedom of the problem matches the total number of the expansion coefficients. Displacements and rotations are computed for noncylindrical helical springs as well as cylindrical helical springs, and parameters that affect the convergence of the solution are discussed. This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han Jinhee Lee received B. S. and M. S. degrees from Seoul National Uni-versity and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from University of Michigan in 1992 and joined Dept. of Mechano-Informatics of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Free Oscillations of a Composite Structurally Orthotropic Cylindrical Shell Stiffened by Discretely Positioned Ring Ribs

A method for calculating the natural oscillations of a cylindrical shell of an orthotropic material is proposed. The shell is stiffened by a set of sufficiently densely positioned transverse-longitudinal ribs, the arrangement of which allows “smearing” and contains sparsely positioned discrete stiffening rings. The shell may have a closed or open cross section; it is considered to be loaded by external pressure and axial forces. The problem is reduced to a set of homogeneous algebraic equations the number of which is equal to twice the number of discrete ribs; the equation was obtained in explicit form. The comparison of the calculated and experimental data is provided.     

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.     

Effect of methods for fixing and loading of cylindrical shells on their stability and supercritical behavior

The effect of the methods of fixing and external-pressure loading of an elastic isotropic cylindrical shell on its sub- and supercritical deformation is investigated. A geometrically nonlinear statement of the edge problem in the form of the technical theory of finite deflection hollow shells is applied. The edge problem is digitized with the Rayleigh-Ritz method. The solution to the set of nonlinear algebraic equations is sought using the methods of continuation by the parameter close to the optimal one. The versions of shell sealing and supporting as well as uniform lateral pressure and uniform compression are considered. The trajectories of shell loading are plotted and the shapes of their supercritical equilibrium states are found. The axial compressing load is found to exert a larger effect on the upper and lower critical pressure as compared to the conditions of shell end fixing. Moreover, axial loading of short shells yields an increase in the critical pressure rather than its decrease, as is customary in the theory of shell stability.     

Pseudospectral analysis of buckling and free vibration of rectangular Mindlin plates

A study of buckling and free vibration of rectangular Mindlin plates is presented. The analysis is based on the pseudospectral method, which uses basis functions that satisfy the boundary conditions. The equations of motion are collocated to yield a set of algebraic equations that are solved for the critical buckling load and for the natural frequencies in the presence of the in-plane loads. Numerical examples of rectangular plates with SS-C-SS-C boundary conditions are provided for various aspect ratios and thickness ratios, which show good agreement with those of the classical plate theory when the thickness ratio is very small. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from the University of Michigan, Ann Arbor in 1992 and joined the Dept. of Mechanical and Design Engineering of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

受多点横向非定常约束梁的稳态与瞬态响应

研究受有多点横向非定常约束梁的动态响应问题。利用横向非定常约束为周期函数梁的微分代数方程的线性特点,确定出多基频周期性非定常约束激励下梁稳态响应的解析解;利用截断模态的微分代数方程确定出受有横向非定常约束梁的瞬态响应。为此,首先利用无阻尼截断模态微分代数方程的齐次形式,将欧拉-伯努利梁的等效多跨梁模态表示为简单边界条件下模态函数的线性组合,再利用得到的模态函数与位移影响函数重新将微分代数方程表示为常微分方程的形式,进而得到横向非定常约束作用下梁的瞬态响应解的积分形式。在此过程中,研究了微分代数方程齐次式的特征值及其数量、求解方法等相关问题。通过单点多频率的横向非定常约束作用的梁及多点单频横向非定常约束作用梁的算例,重点分析了非定常约束位置与基频对梁稳态响应的影响。结果表明：在欠阻尼下,两种梁模态响应的极大值均在等效多跨梁的各个主频附近;单点横向非定常约束作用响应极小值在简单边界条件梁的各个主频附近,而多点横向非定常约束作用梁响应极小值的分布比较复杂。算例也说明了所提出的方法是正确、有效的。     

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.     

Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations

Approximate solutions for the bending of moderately thick rectangular plates on two-parameter elastic foundations (Pasternak-type) as described by Mindlin's theory are presented. The plates are subjected to an arbitrary combination of clamped and simply-supported boundary conditions. An efficient computational technique, the differential quadrature (DQ) method, is employed to transform the governing differential equations and boundary conditions into a set of linear algebraic equations for approximate solutions. These resulting algebraic equations are solved numerically. In this study, the accuracy of the DQ method is established by direct comparison with results in the existing literature. The convergence properties of the method are illustrated for different combinations of boundary conditions. The deflections, moments and shear forces at selected locations are tabulated in detail for different elastic foundations. The efficiency and simplicity of the solution method are highlighted.     

Few-view gamma tomography used to monitor scabbing and shear fracture in a spherical iron shell compressed by explosion

The effectiveness of few-view gamma tomography for nondestructive testing of the internal structure of scabbing and shear fracture in a spherical iron shell subjected to explosive compression has been studied experimentally. To obtain shadow X-ray radiograms of the shell, a fine-focus betatron and a screened photographic film are used. The results of reconstruction performed by applying a modified multiplicative algebraic reconstruction technique and subsequent postprocessing of tomograms of a meridional section of the compressed shell are considered. It is shown that the precision of reconstruction of the spatial pattern of fractures mostly depends on the number of projections used during the experiment.     

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.     

脉冲激励振动问题的高精度数值分析

利用数值稳定性好、计算精度高的重心Lagrange插值近似未知函数,得到未知函数各阶导数的微分矩阵,提出高精度数值分析任意激励下振动问题的重心插值配点法.采用附加法施加初始条件,得到一个n+2个方程n个未知量的代数方程组.利用最小二乘法求解线性方程组,得到振动位移,进而利用微分矩阵直接求得振动的速度和加速度.对于脉冲激励等复杂激励下的振动问题,将前一个时间区域末的位移和速度作为下一个时间区域的初始条件,分段进行数值分析.数值算例表明重心插值配点法在分析脉冲激励下的振动问题,具有非常高的计算精度.     

Analysis of annular Reissner/Mindlin plates using differential quadrature method

The axisymmetric flexure responses of moderately thick annular plates under static loading are investigated. The shear deformation is considered using the first-order Reissner/Mindlin plate theory and the solutions are obtained using the differential quadrature (DQ) method. In the solution process, the governing differential equations and boundary conditions for the problem are initially discretized by the DQ algorithm into a set of linear algebraic equations. The solutions of the problem are then determined by solving the set of algebraic equations. This study considers the plate subjected to various combinations of clamped, simply-supported, free and guided boundary conditions and different loading manners. The accuracy of the method is demonstrated through direct comparison of the present results with the corresponding exact solutions available in the literature.     

Thermal analysis of viscoelastic multibody systems

Thermal analysis of the transient motion of a multibody system, consisting of interconnected rigid, elastic, and viscoelastic components is presented. The configuration of each elastic or viscoelastic body is identified using three sets of modes, rigid-body, reference and normal modes. Rigid-body modes describe translations and large rotations of the body reference. The finite element method is employed to describe body deformations, and reference conditions are imposed in order to define a unique displacement field. Accordingly, normal modes of vibrations of flexible components can be used to reduce the number of elastic coordinates, thus eliminating high-frequency modes of vibration. The system equations of motion are formulated using Lagrange's equation, and nonlinear algebraic equations, describing constraints between adjacent bodies, are coupled to the equations of motion using a Lagrange multiplier technique.     

各种板边条件下大挠度圆板自由振动的分岔解

计及几何非线性,在各种板边条件下,建立圆板自由振动的非线性动力学方程.采用Galerkin法,将圆板的非线性动力学偏微分方程简化成三种标准类型的Duffing方程.提出一类强非线性动力系统的两项谐波法,将描述动力系统的二阶常微分方程,化为以频率、振幅为变量的非线性代数方程组,考虑初始条件补充约束方程,构成频率、振幅为变量的封闭非线性代数方程组.利用Maple程序可以方便地求解.结果表明,两项谐波法不仅适合于对称振动问题,而且适合于非对称振动问题.     

矩形弹性壳液耦合系统的非线性振动分析

从流体运动微分方程组和边界条件出发,利用拉格朗日方程,建立矩形弹性壳液耦合系统的非线性振动方程组,并进行数学仿真计算,得到产生稳定的低频大幅重力波的条件及重力波和壳的幅频曲线,数值计算结果与实验现象吻合较好.     

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.     

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.     

Thermohydrodynamic Analysis and Design of Dual Conical-Cylindrical Bearings Using Non-Newtonian Lubricants

In this study, further analysis and design of the hydraulic servo cylinder using dual conical-cylindrical bearings with practical dimensions are considered, and the dimensional forms can be used to express the bearing performance parameters. The film viscosity is formulated as an exponential function of temperatures and pressures. The power law, the non-Newtonian lubricant, the modified Reynolds equation, and thermohydrodynamic (THD) boundary conditions are the basic assumptions of this analysis. Effects of the eccentricity and tilt angle are considered herein.