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。      

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.     

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.     

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.     

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.     

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 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.     

TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem (K−ω²M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick-Williams algorithm which gives the number of natural frequencies below ω_t, a trial value of ω, as ∑J_m+s{K (ω_t)} wheres {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped-clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick-Williams algorithm. The three solution forms are: ∣K (ω)∣=0; D=0 with ∣K (ω)∣→∞; and ∣K (ω)∣≠0 with D≠0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.     

Wave based analysis of the Green's function for a layered cylindrical shell

Cylindrical shells composed of concentric layers may be designed to affect the way that elastic waves are generated and propagated, particularly when some layers are anisotropic. To aid the design process, the present work develops a wave based analysis of the Green's function for a layered cylindrical shell in which the response is given as a sum of waves propagating in the axial coordinate. The analysis assumes linear Hookean materials for each layer. It uses finite element discretizations in the radial coordinate and Fourier series expansions in the circumferential coordinate, leading to linear equations in the axial wavenumber domain that relate shell displacements and forces. Inversion to the axial domain is accomplished via a state-space formulation that is evaluated using residue integration. The resulting expression for the Green's function for each circumferential harmonic is a summation over the natural waves of the shell. The finite element discretization in the radial direction allows the approach to be used for arbitrarily thick shells. The approach is benchmarked to results from an isotropic shell and numerical examples are given for a shell composed of a fiber-reinforced material. The numerical examples illustrate the effect of fiber orientation on the Green's function.     

A SEMI-ANALYTICAL COUPLED FINITE ELEMENT FORMULATION FOR COMPOSITE SHELLS CONVEYING FLUIDS

A coupled formulation based on the semi-analytical finite element technique is developed for composite shells conveying fluid. The structural finite element formulation is from Ramasamy and Ganesan (1998 Computers and Structures70, 363-376), while the fluid part is modelled by the characteristic wave equation. The fluid part is modelled using a velocity potential formulation and the dynamic pressure acting on the walls is derived from Bernoulli's equation. Impermeability and dynamic condition are imposed on the fluid-structure interface. The finite element equations for the composite shell conveying fluid are validated using available results in the literature. A detailed parametric study is carried out for various boundary conditions as well as for different length-to-radius and radius-to-thickness ratios.     

DYNAMICS AND DISTRIBUTED CONTROL OF CONICAL SHELLS LAMINATED WITH FULL AND DIAGONAL ACTUATORS

Nozzles, rocket fairings and many engineering structures/components are often made of conical shells. This paper focuses on the finite element modelling, analysis, and control of conical shells laminated with distributed actuators. Electromechanical constitutive equations and governing equations of a generic piezo(electric)elastic continuum are defined first, followed by the strain-displacement relations and electric field-potential relations of laminated shell composites. Finite element formulation of a piezoelastic shell element with non-constant Lamé parameters is briefly reviewed; element and system matrix equations of the piezoelastic shell sensor/actuator/structure laminate are derived. The system equation reveals the coupling of mechanical and electric fields, in which the electric force vector is often used in distributed control of shells. Finite element eigenvalue solutions of conical shells are compared with published numerical results first. Distributed control of the conical shell laminated with piezoelectric shell actuators is investigated and control effects of three actuator configurations are evaluated.     

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.     

Free vibration of non-circular cylindrical shells with longitudinal interior partitions

An analysis is presented for the free vibration of a simply supported non-circular cylindrical shell with longitudinal interior partitions. For this purpose, the governing equations of vibration of a non-circular cylindrical shell including a plate as special case are written in a matrix differential equation by using the transfer matrix in the circumferential direction. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrix of the shell and the point matrix at the joint of the structure matrix. The method is applied to approximately elliptical cylindrical shells with an interior plate, and the natural frequencies and the mode shapes of vibration are calculated numerically, giving the results.     

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes (n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.     

Numerical analysis of immersed finite cylindrical shells using a coupled BEM/FEM and spatial spectrum approach

This study numerically analyzes submerged cylindrical shells using a coupled boundary element method (BEM) with finite element method (FEM) in conjunction with the wave number theory, in which the spatial Fourier transform of surface velocity for cylinders is directly related to pressure in a far field. The acoustic loading is formulated using a symmetric complex matrix derived from a boundary integral equation where the symmetry is based on an acoustic reciprocal principle for surface acoustics. In this formulation the acoustic loading matrix is a large acoustic element whose degree of freedom is connected to the normal displacement of the vibrating structures. The coupled BEM/FEM equation is a banded, symmetric matrix, and thus its bandwidth can be minimized using a proper algorithm. This formulation significantly increases numerical efficiency. The computed normal velocity is thus transformed to wave number representation to examine acoustic radiation. A finite plane cylindrical shell, without attached stiffeners, and a shell with internal ring stiffeners are chosen to demonstrate the present analysis procedure. The far field pressure computed directly from the integral equation and predicted by wave number theory correlates closely with increasing vibrating frequency. Meanwhile, the influences of the internal ring structures on acoustic radiation are examined using the wave number theory, which helps in understanding how internal structures influence radiated noise.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

Free vibration of a thin cylindrical shell with periodic circumferential stiffeners

The theory is developed for obtaining the propagation constants of a thin uniform cylindrical shell, periodically stiffened by uniform circular frames of general cross-section. The free wave motion is analyzed and the stop and pass bands of free wave motion in the structure are located. Hysteretic damping is included. The natural frequencies of two stiffened finite cylindrical shells are deduced. The relative effects of the frame cross section and pitch on the free vibration characteristics of the whole structure are discussed.     

Mechanics and finite elements for the damped dynamic characteristics of curvilinear laminates and composite shell structures

Integrated mechanics and a finite element method are presented for predicting the damping of doubly curved laminates and laminated shell composite structures. Damping mechanics are formulated in curvilinear co-ordinates from ply to structural level and the structural modal loss factors are calculated using the energy dissipation method. The modelling of damping at the laminate level is based on first order shear shell theory. An eight-node shell damping finite element is developed. Comparisons of the present model with classical and discrete layer laminate damping theory predictions are shown. Modal damping and natural frequencies of composite plates and open cylindrical shells were measured and correlated with predicted results. Parametric studies illustrate the effect of curvature and lamination on modal damping and natural frequency.     

Free vibrations of laminated composite cylindrical shells with an interior rectangular plate

The free vibration analysis of a laminated composite cylindrical shell with an interior rectangular plate is performed by the analytical and experimental methods. The frequency equations of vibration of the shell including the plate are formulated by using the receptance method. To obtain the free vibration characteristics before the combination of two structures, the energy principle based on the classical plate theory and Love's thin shell theory is adopted. The numerical results are compared with the results from an experiment, as well as a finite element analysis, to validate the current formulation. The influences of the length-to-radius ratio (L_S/a) and radius-to-thickness ratio (a/h_S) of the shell and fiber orientation angles (Θ) of symmetric cross- and angle-ply composite materials on the natural frequencies of a cylindrical laminated combined shell are also discussed in details.     

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.