Method of Green’s function of nonlinear vibration of corrugated shallow shells

Based on the dynamic equations of nonlinear large deflection of axisymmetric shallow shells of revolution, the nonlinear free vibration and forced vibration of a corrugated shallow shell under concentrated load acting at the center have been investigated. The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green’s function. To solve the integral-differential equations, the expansion method was used to obtain Green’s function. Then the integral-differential equations were reduced to the form with a degenerate core by expanding Green’s function as a series of characteristic function. Therefore, the integral-differential equations became nonlinear ordinary differential equations with regard to time. The amplitude-frequency relation, with respect to the natural frequency of the lowest order and the amplitude-frequency response under harmonic force, were obtained by considering single mode vibration. As a numerical example, nonlinear free and forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied. The obtained solutions are available for reference to the design of corrugated shells.     

DYNAMICS BEHAVIOR OF AXISYMMETRIC AND BEAM-LIKE ANISOTROPIC CYLINDRICAL SHELLS CONVEYING FLUID

The flow-induced vibration characteristics of anisotropic laminated cylindrical shells partially or completely filled with liquid or subjected to a flowing fluid are studied in this work for two cases of circumferential wave number, the axisymmetric, where n=0 and the beam-like, where n=1. The shear deformation effects are taken into account in this theory; therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The present method is a combination of finite element analysis and refined shell theory in which the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear co-ordinates. Mass and stiffness matrices are determined by precise analytical integration. A finite element is defined for the liquid in cases of potential flow that yields three forces (inertial, centrifugal and Coriolis) of moving fluid. The mass, stiffness and damping matrices due to the fluid effect are obtained by an analytical integration of the fluid pressure over the liquid element. The available solution based on Sanders' theory can also be obtained from the present theory in the limiting case of infinite stiffness in transverse shear. The natural frequencies of isotropic and anisotropic cylindrical shells that are empty, partially or completely filled with liquid as well as subjected to a flowing fluid, are given. When these results are compared with corresponding results obtained using existing theories, very good agreement is obtained.     

Maxi-element analysis for free vibrations of orthotropic shells of revolution

The theory of free vibration for orthotropic shells of revolution with arbitrary homogeneous boundary conditions is developed. The essence of the method is to decompose the overall shell into a number of so-called cylindrical, conical, and plate “maxi-elements”. Since the eigenfunctions of each of the individual maxi-elements are analytically determined directly from the solution of the governing differential equations, the procedure has the advantage of requiring significantly fewer elements compared with the usual finite element or finite difference procedures. For the conical shell and the plate, the method of solution is novel, while the solution for the cylindrical shell has been published by the first author. The versatility and accuracy of the method is shown through the inclusion of a number of examples which present the excellent correlation with test results and other numerical schemes.     

Free vibration of non-circular cylindrical shells with variable circumferential profile

An analysis is presented of the free vibration of non-circular cylindrical shells with a variable circumferential profile expressed as an arbitrary function. The applicability of thin-shell theory is assumed and the governing equations of vibration of a non-circular cylindrical shell are written in a matrix differential equation by using the transfer matrix of the shell. Once the transfer matrix has been determined by numerical integration of the matrix equation, the natural frequencies and mode shapes of vibration are calculated numerically in terms of the matrix elements. The method is applied to cylindrical shells of three or four-lobed cross-section, and the effects of the length of the shell and the radius at the lobed corners on the vibration are studied.     

扬声器辐射体旋转薄壳的非线性振动方程

推导了扬声器辐射体旋转薄壳的离散非线性振动方程。从虚功原理出发,选用扬声器辐射体旋转薄壳的本征模态对连续体进行离散。薄壳的几何非线性采用Sanders非线性薄壳理论的应变一位移关系。方程系数由有限元方法确定。方程表明轴对称模态由驱动力直接激励,非轴对称模态由轴对称模态非线性耦合激励,该耦合激励表现为参数激励。方程揭示了扬声器非线性失真的机制,可用于分析扬声器辐射体薄壳非线性引起的谐波失真、分谐波失真和互调失真。      

一种弯张换能器及其共振频率计算方法

提出一种弯张换能器即欧米伽换能器,推导出其共振频率和位移振形函数。把欧米伽换能器分成四个构成部分,利用旋转薄壳理论和压电理论分别求出各部分的能量并进行相加,得到整个欧米伽换能器能量的泛函表达式;把几何边界连续条件和包含待定系数的位移振形函数代入到欧米伽换能器能量泛函中,运用Rayleigh-Ritz法求出欧米伽换能器的共振频率,再把共振频率代入Rayleigh-Ritz偏微分方程和边界方程中求出位移振形函数的待定系数以获得确定的位移振形函数。该方法也被推广到对钹式换能器共振频率和位移振形函数的求解上。上述求解结果与实验结果和数值软件相结合论证了该方法的有效性。可获得以下结论:(1)欧米伽换能器陶瓷片的径向振动与金属端帽顶部的纵向振动同相,减少了声场中的反相分量,易作为低频换能器使用;(2)为解决欧米伽换能器和钹式换能器的优化设计提供了理论支持;(3)文中求解共振频率和位移振形函数的方法,即可以应用在由旋转薄壳组成的弯张换能器上也可以应用在由旋转薄壳组成的其它结构上,具有普遍性。      

Free vibrations of laminated composite shells with uniformly distributed attached mass using higher order shell theory including stiffness effect

In this paper, free vibrations of a cross-ply composite shell with or without a uniformly distributed attached mass are analyzed using higher order shell theory. The results of free vibrations without distributed attached mass are validated by previous literatures. The stiffness effect of this distributed attached mass are also considered and compared with those well-known published results in which this effect is ignored. Various results for composite shells under a variety of conditions such as variations in the thickness of the shell, variation in the thickness of the distributed attached mass, variation in the radii of curvatures and various elasticity moduli are presented in this paper. In some cases, to verify the novel results, first-order shear deformation theory (FSDT) is also used. In this paper, parameters which influence the natural frequencies of the shells with attached mass including the stiffness of the mass are investigated. Parameters which are investigated in this paper are thickness of the shell, thickness of the distributed attached mass, elasticity moduli of the distributed attached mass and radius of curvatures of shells. Increasing the thickness or elasticity moduli of the distributed attached mass will increase the fundamental natural frequency of the shell. The effect of the stiffness of the distributed attached mass is decreased by decreasing the radii of curvatures or increasing the thickness of the shells.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

考虑铺层角的复合材料圆柱壳自由振动三维弹性准确解

为了获得任意角度铺层的多层复合材料圆柱壳的自由振动准确解,在三维弹性理论的基础上,结合分层理论和状态空间法,建立横向位移和应力的传递矩阵,轴向和环向位移采用双螺旋模式的位移函数,对任意角度铺层复合材料圆柱壳简支边界条件下的自由振动进行了理论推导,得到了自由振动方程的精确形式。与文献理论解和有限元计算结果对比,结果表明,关注频率在2倍的环频率以下时,薄壳的固有频率计算精度能控制在1%以内,厚壳的固有频率计算精度能控制在2%以内。对于厚壳的计算可将壳体沿厚度方向划分为多层来处理,这样能有效提高计算精度。计算分析了铺层角对壳体固有频率的影响,环向模态数较低时,固有频率随着铺层角的增加呈抛物线变化趋势;环向模态数较高时,固有频率随着铺层角的增大单调递增。该理论方法同样适用于均质各向同性壳和正交各向异性圆柱壳。      

Influence of initial geometrical imperfections on vibrations of axially compressed stiffened cylindrical shells

The vibrations of stiffened cylindrical shells having axisymmetric or asymmetric initial geometrical imperfections and axial preload are analyzed. The analysis is based on a solution of the von Kárman-Donnell non-linear shell equations, an “exact” solution of the compatibility equation, and a first order approximation by the Galerkin method of the equilibrium equation. The stiffeners are closely spaced and “smeared” stiffener theory is employed. The results of an extensive parametric study carried out on shells similar to those used in vibration and buckling tests at the Technion show that stiffening of the shell will lower the imperfection-sensitivity of its free vibrations, but the decrease depends on the type of stiffening (stringers or rings), the mode shapes of the vibration and the imperfection, the stiffener strength and eccentricity. The imperfection-sensitivity decrease, caused by the stiffeners, is greater for vibration mode shapes with high imperfection-sensitivity than for other vibration mode shapes. The sensitivity differences between stringer and ring-stiffened shells depend especially on the vibration and the imperfection mode shapes, and on their coupling. Small imperfections change the natural frequencies of stiffened shells in the same directions as for isotropic shells, but to a smaller extent. The frequency dependence on the external load is also strongly affected by the imperfection mode shape. The results correlate well with earlier ones for isotropic shells.     

Numerical perturbation solutions for the vibrations of prestressed,clamped cylindrical shells

A perturbation technique is used to reduce the eighth-order vibration problem for prestressed, clamped cylindrical shells to an equivalent sixth-order membrane problem. In the transformation to a membrane problem composite expansions are utilized, uniformly valid over the length of the shell, to formulate modified boundary conditions that account for the effects of bending near the shell ends. By solving the simpler modified membrane problem numerically, one can demonstrate the effectiveness of the method against eighth-order bending solutions. Indeed, the distinguishing characteristics of the proposed technique is the manner in which perturbation theory and numerical analysis methods complement one another as, for example, in the case of the finite element method, where under certain conditions a modification of the simple membrane element would extend the inherent numerical efficiency of the element to the solution of a class of problems involving both membrane and bending actions.     

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.     

Free vibration of joined conical-cylindrical shells

An analysis is presented for the free vibration of joined conical-cylindrical shells. The governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method has been applied to a joined truncated conical-cylindrical shell and an annular plate-cylindrical shell system, and the natural frequencies and the mode shapes of vibration calculated numerically. The results are presented.     

Wave propagation characteristics of helically orthotropic cylindrical shells and resonance emergence in scattered acoustic field. Part 1. Formulations

The method of wave function expansion is adopted to study the three dimensional scattering of a plane progressive harmonic acoustic wave incident upon an arbitrarily thick-walled helically filament-wound composite cylindrical shell submerged in and filled with compressible ideal fluids. An approximate laminate model in the context of the so-called state-space formulation is employed for the construction of T-matrix solution to solve for the unknown modal scattering coefficients. Considering the nonaxisymmetric wave propagation phenomenon in anisotropic cylindrical components and following the resonance scattering theory which determines the resonance and background scattering fields, the stimulated resonance frequencies of the shell are isolated and classified due to their fundamental mode of excitation, overtone and style of propagation along the cylindrical axis (i.e., clockwise or anticlockwise propagation around the shell) and are identified as the helically circumnavigating waves.     

Natural vibrations and stability of shells of revolution interacting with an internal fluid flow

A finite-element algorithm is proposed to investigate the dynamic behavior of elastic shells of revolution containing a quiescent or a flowing inviscid fluid in the framework of linear theory. The fluid behavior is described using the perturbed velocity potential. The shell behavior is treated in the framework of the classical shell theory and variational principle of virtual displacements incorporating a linearized Bernoulli equation for calculation of hydrodynamic pressure acting on the shell. The problem reduces to evaluation and analysis of the eigenvalues in the connected system of equations obtained by coupling the equations for velocity perturbations with the equations for shell displacements. For cylindrical shells, the results of numerical simulations are compared with recently published experimental, analytical and numerical data. The paper also reports the results of studying the dynamic behavior of shells under various boundary conditions for the perturbed velocity potential. The investigation made for conical shells has shown that under certain conditions an increase in the cone angle can change a divergent type of instability to a flutter type.     

Structural response of cylindrically curved laminated composite shells subjected to blast loading

This paper is concerned with the theoretical analysis and correlation with the numerical results of the displacement time histories of the cylindrically curved laminated composite shells exposed to normal blast shock waves. The laminated composite shell is clamped at its all edges. The dynamic equation of the cylindrical shell used in this study is valid under the assumptions made in Love's theory of thin elastic shells. The constitutive equations of laminated composite shells are given in the frame of effective modulus theory. The governing equation of the cylindrical shell is solved by the Runge-Kutta method. In addition, a finite element modeling and analysis are presented and compared with the theoretical results. The peak deflections and response frequencies obtained from theoretical and numerical analyses are in agreement. The effects of material properties and geometrical properties are examined on the dynamic behaviour.     

Free vibration of a conical shell with variable thickness

An analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach. The applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined by quadrature of the equations, the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges. The method is applied to truncated conical shells with linearly, parabolically or exponentially varying thickness, and the effects of the semi-vertex angle, truncated length and varying thickness on the vibration are studied.     

Vibration analysis of laminated cross-ply oval cylindrical shells

Here, free vibrations and transient dynamic response analyses of laminated cross-ply oval cylindrical shells are carried out. The formulation is based on higher order theory that accounts for the transverse shear and the transverse normal deformations, and includes zig-zag variation in the in-plane displacements across the thickness of the multi-layered shells. The contributions of inertia effect due to in-plane and rotary motions, and the higher order function arising from the assumed displacement models are included. The governing equations obtained using Lagrangian equations of motion are solved through finite element approach. A detailed parametric study is conducted to bring out the influence of different shell geometry, ovality parameter, lay-up and loading environment on the vibration characteristics related to different modes of vibrations of oval shell.     

Dynamic analysis of layered composite shells using nine node degenerate shell elements

The basic objective of the work reported in this paper is to extend a nine-node degenerated shell element developed earlier for stress analysis to the free vibration analysis of thick laminated composites. The nine-noded degenerated shell element is preferable to conventional solid elements for the modeling and analysis of laminated composite shell structures since the shell element works for both thick and thin shells. An enhanced interpolation of the transverse shear strains in the natural coordinates is used in this formulation to produce a shear locking free element and an enhanced interpolation of the membrane strains in the local coordinates is used to produce a membrane locking free element. The interpolation functions used in formulating the assumed strains are based on the Lagrangian interpolation polynomials. Various numerical examples are analyzed and their results are compared with the existing exact solutions where available and the numerical solutions calculated from other shell finite element formulations, to benchmark the current formulation.     

Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory

Based on the modified couple-stress theory, three-dimensional analytical solutions of free vibration of a simply supported, multilayered and anisotropic composite nanoplate are derived by solving an eigenvalue system and using the propagator matrix method. By expanding the solutions of the extended displacements in terms of two-dimensional Fourier series, the final governing equations of motion with modified couple-stress effect are reduced to an eigenvalue system of ordinary differential equations. Analytical expressions for the natural frequencies and mode shapes of multilayered anisotropic composite plates with modified couple-stress effect are then derived via the propagator matrix method. Numerical examples are carried out for homogeneous thick-plates and sandwich composite plates to show the effect of the non-local parameter in different layers and stacking sequence on the mode shapes. The present solutions can serve as benchmarks to various thick-plate theories and numerical methods, and could be further useful for designing layered composite structures involving small scale.