大跨度双层网壳的非线性动态响应

网壳结构在大跨度结构中得到广泛应用．在建立了双层网格扁球壳的非线性强迫振动微分方程的基础上,研究了在边缘滑动固定的边界条件下,双层网格扁球壳的非线性动态响应问题．用突变理论建立了该网壳的尖点突变模型,得出了突变的临界方程,并阐述了网壳参数对该结构动态屈曲的影响．     

Nonlinear modes of cylindrical panels with complex boundaries. <Emphasis Type="Italic">R</Emphasis>-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.     

Nonobvious features of dynamics of circular cylindrical shells

In the framework of the nonlinear theory of flexible shallow shells, we study free bending vibrations of a thin-walled circular cylindrical shell hinged at the end faces. The finite-dimensional shell model assumes that the excitation of large-amplitude bending vibrations inevitably results in the appearance of radial vibrations of the shell. The modal equations are obtained by the Bubnov-Galerkin method. The periodic solutions are found by the Krylov-Bogolyubov method. We show that if the tangential boundary conditions are satisfied “in the mean,” then, for a shell of finite length, significant errors arise in determining its nonlinear dynamic characteristics. We prove that small initial irregularities split the bending frequency spectrum, the basic frequency being smaller than in the case of an ideal shell.     

Nonlinear Modes of Motion of Thin Circular Cylindrical Shells

Free flexural vibrations of a simply supported shell are studied within the framework of the nonlinear theory of flexible shallow shells. It is assumed that largeamplitude flexural vibrations are coupled with radial vibrations of the shell. Modal equations are derived by the Bubnov–Galerkin method. Periodic solutions are obtained by the Krylov–Bogolyubov method. The skeleton curve of the soft type obtained using a nonlinear finitedimensional shell model agrees with available experimental data.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

On effect of initial imperfections on parametric vibrations of cylindrical shells with geometrical non-linearity

The effect of initial imperfections on the parametric vibrations of cylindrical shells is analyzed. The shell has moderate amplitudes of vibrations; therefore, geometrically nonlinear theory is used. The shell vibrations are described by the Donnel equations. The interaction of three pairs of conjugate modes is considered in the analysis. Therefore, the shell vibrations are described by six-degrees-of-freedoms nonlinear dynamical system. The multiple scales method and the continuation technique are used to analyze the system dynamics. The role of initial imperfections in nonlinear dynamics of shell is discussed using frequency responses.     

Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

Non-linear vibrations of imperfect free-edge circular plates and shells

Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Kármán equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.     

Chaotic Vibrations of a Cylindrical Shell-Panel with an In-Plane Elastic-Support at Boundary

This paper presents numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation. The shell, with a rectangular boundary, is simply supported for deflection and the shell is constrained elastically in an in-plane direction. Using the Donnell--Mushtari--Vlasov equation, modified with an inertia force, the basic equation is reduced to a nonlinear differential equation of a multiple-degree-of-freedom system by the Galerkin procedure. To estimate regions of the chaos, first, nonlinear responses of steady state vibration are calculated by the harmonic balance method. Next, time progresses of the chaotic response are obtained numerically by the Runge--Kutta--Gill method. The chaos accompanied with a dynamic snap-through of the shell is identified both by the Lyapunov exponent and the Poincaré projection onto the phase space. The Lyapunov dimension is carefully examined by increasing the assumed modes of vibration. The effects of the in-plane elastic constraint on the chaos of the shell are discussed.     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

Free vibrations of circular cylindrical shells with a small added concentrated mass

The effect of a small added mass on the frequency and shape of free vibrations of a thin shell is studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial flexural vibrations. The interaction of shape-generating waves is studied using modal equations obtained by the Bubnov–Galerkin method. Splitting of the flexural frequency spectrum is found, which is caused not only by the added mass but also by the wave-formation parameters of the shell. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of flexural and radial vibrations can be neglected.     

正弦波纹扁球壳的非线性强迫振动

波纹壳是传感器弹性元件的一类重要形式,也是精密仪器仪表弹性元件中的一类重要形式。由于波纹壳形状复杂、参数众多、厚度薄,对其进行非线性分析非常重要同时也是十分困难的。本文考虑一种在传感器弹性元件中有重要应用价值的正弦波纹浅球壳体,将这种壳体视为结构上的圆柱正交异性扁球壳,根据Andryewa的思想,分别得到了正弦波纹壳径向、环向在拉伸、弯曲下的等价的四个各向异性参数;建立了正弦波纹扁球壳的非线性强迫振动微分方程;得到了正弦波纹扁球壳非线性强迫振动的共振周期解及幅频特性曲线。     

The Problem of Forced Nonlinear Vibrations of Cylindrical Shells Completely Filled with Liquid

The forced nonlinear vibrations of a thin cylindrical shell completely filled with a liquid are studied. A refined mathematical model is used. The model takes into account the nonlinear terms up to the fifth power of the generalized displacement of the shell. The Bogolyubov’Mitropolsky averaging method is used to plot amplitude’frequency response curves for steady-state vibrations. The steady-state vibrations at the frequency of principal harmonic resonance are analyzed for stability__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 52–59, February 2005.     

Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries

In the present study, the large-amplitude vibrations and stability of a perfect circular cylindrical shell subjected to axial harmonic excitation in the neighborhood of the lowest natural frequencies are investigated. Donnell's shallow shell theory is used and the shell spatial discretization is obtained by the Ritz method. An efficient low-dimensional model presented in previous publications is used to discretize the continuous system. The main purpose of this work is to discuss the use of basins of attraction as a measure of the reliability and safety of the structure. First, the nonlinear behavior of the conservative system is discussed and the basin structure and volume is understood from the topologic structure of the total energy and its evolution as a function of the system parameters. Then, the behavior of the forced oscillations of the harmonically excited shell is analyzed. First the stability boundaries in force control space are obtained and the bifurcation events connected with these boundaries are identified. Based on the bifurcation diagrams, the probability of parametric instability and escape are analyzed through the evolution and erosion of basin boundaries within a prescribed control volume defined by the manifolds. Usually, basin boundaries become fractal. This together with the presence of catastrophic subcritical bifurcations makes the shell very sensitive to initial conditions, uncertainties in system parameters, and initial imperfections. Results show that the analysis of the evolution of safe basins and the derivation of appropriate measures of their robustness is an essential step in the derivation of safe design procedures for multiwell systems.     

A closed-form solution procedure for circular cylindrical shell vibrations

A systematic procedure for obtaining the closed-form eigensolution for thin circular cylindrical shell vibrations is presented, which utilizes the computational power of existing commercial software packages. For cylindrical shells, the longitudinal, radial, and circumferential displacements are all coupled with each other due to Poissons ratio and the curvature of the shell. For beam and plate vibrations, the eigensolution can often be found without knowledge of absolute dimensions or material properties. For cylindrical shell vibrations, however, one must know the relative ratios between shell radius, length, and thickness, as well as Poissons ratio of the material. The mode shapes and natural frequencies can be determined analytically to within numerically determined coefficients for a wide variety of boundary conditions, including elastic and rigid ring stiffeners at the boundaries. Excellent agreement is obtained when the computed natural frequencies are compared with known experimental results.     

Nonlinear Oscillations and Stability of Parametrically Excited Cylindrical Shells

Based on Donnell shallow shell equations, the nonlinear vibrations and dynamic instability of axially loaded circular cylindrical shells under both static and harmonic forces is theoretically analyzed. First the problem is reduced to a finite degree-of-freedom one by using the Galerkin method; then the resulting set of coupled nonlinear ordinary differential equations of motion are solved by the Runge–Kutta method. To study the nonlinear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, Lyapunov exponents, stable and unstable fixed points, bifurcation diagrams, and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric excitation of flexural modes and escape from the pre-buckling potential well. Calculations are carried out for the principal and secondary instability regions associated with the lowest natural frequency of the shell. Special attention is given to the determination of the instability boundaries in control space and the identification of the bifurcational events connected with these boundaries. The results clarify the importance of modal coupling in the post-buckling solution and the strong role of nonlinearities on the dynamics of cylindrical shells.     

横观各向同性圆球壳的自由扭转振动

本文用弹性动力学理论研究横观各向同性圆球壳的轴对称自由扭转振动问题,求出位移和应力的解析表达式,揭示了壳体在子午线方向和半径方向的耦合振动特性,文末算例给出不同几何尺寸和材料性质圆球壳固有频率和振型的数字计算结果.     

Vibrations of structurally orthotropic laminated shells under thermal power loading

On the basis of the linearized version of equations obtained in a geometrically nonlinear statement and describing the nonaxisymmetric strain of nonshallow sandwich structure orthotropic shells under thermal power loading, the Rayleigh–Ritz method with polynomial approximation of displacements and shear strains is used to solve the problem of small free vibrations of axisymmetrically thermally preloaded freely supported three-layer conical shell. The causes of dynamical fracture of the shell under study are revealed.