The Nonlinear Vibrations and Stability of Orthotropic Cylindrical Liquid-Carrying Shells Under Periodic Actions

The nonlinear resonance vibrations and stability of nonlinear orthotropic shells partially filled with a liquid and subjected to longitudinal and transverse periodic loading are studied. The equations of motion are derived with regard for the existence and nonlinear interaction of conjugate flexural modes of the shell. Stationary regimes for forced and parametric vibrations are found and analyzed for stability under the conditions of fundamental and subharmonic resonances. The amplitude–frequency characteristics of these regimes are constructed for various parameters of the system     

非惯性参考系中弹性薄板在参数激励与强迫激励联合作用下的1／2亚谐共振分岔分析

本文研究了非惯性参考系中弹性薄板在范围运动与变形运动相互耦合时的1／2亚谐共振分岔,在建立了该系统的动力学控制方程的基础上,利用多尺度法得到了参数激励与强迫激励联合作用下非惯性参考系中弹性薄板1／2亚谐共振时的分岔响应方程及其分岔集,讨论了该动力系统的稳定性,给出了它的五种分响应曲线。     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

超空泡运动体圆柱薄壳的非线性动力屈曲分析

超空泡运动体的动力屈曲失稳具有隐蔽性、突发性和危险性, 因而必须研究清楚运动体的失稳区域边界及失稳振幅. 将超空泡运动体模拟成受轴向周期载荷作用的细长圆柱薄壳, 给出非线性几何方程、物理方程和平衡方程, 建立细长圆柱薄壳带有非线性项的动力屈曲微分方程组; 依据非线性项的形式, 给出合理的非线性位移表达式, 得到具有周期性系数的非线性横向振动微分方程; 采用伽辽金变分法和和鲍洛金方法, 获得带有周期性系数和非线性项的马奇耶方程; 求解非线性马奇耶方程, 得到第一、第二阶不稳定区域内的定态振动振幅的解析表达式; 绘制超空泡运动体的非线性参数共振曲线, 分析航行速度、载荷比例系数、轴向载荷频率和振型对参数共振曲线的影响. 以上研究为建立基于参数共振的圆柱薄壳动力失稳的可靠性分析及基于参数共振可靠性的结构动力优化设计的奠定了理论基础.      

1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions. An erratum to this article is available at .     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

Nonlinear Vibrations of a Cylindrical Shell Containing a Flowing Fluid

The Bogolyubov-Mitropolsky method is used to find approximate periodic solutions to the system of nonlinear equations that describes the large-amplitude vibrations of cylindrical shells interacting with a fluid flow. Three quantitatively different cases are studied: (i) the shell is subject to hydrodynamic pressure and external periodical loading, (ii) the shell executes parametric vibrations due to the pulsation of the fluid velocity, and (iii) the shell experiences both forced and parametric vibrations. For each of these cases, the first-order amplitude-frequency characteristic is derived and stability criteria for stationary vibrations are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 75–84, April 2005.     

Convection of a binary mixture under conditions of thermal diffusion and variable temperature gradient

Instability of a plane horizontal layer of an incompressible binary gas mixture stratified in the gravity field under the action of a transverse temperature gradient modulated in time is studied. The case of solid impermeable boundaries of the layer, where the flux of matter vanishes, is considered. The analysis is based on the Floquet method applied to linearized equations of convection in the Boussinesq approximation. It is shown that there are regions of parametric instability at finite frequencies. In addition to the synchronous or subharmonic response to an external action, the instability may be related to quasiperiodic disturbances. Depending on the amplitude and frequency, modulation can stabilize the unstable basic state and also destabilize the equilibrium of the fluid. The threshold values of convection for modulations of temperature and translational vertical vibrations are compared.     

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love's thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the generalized differential quadrature (GDQ) method. Hill's method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.     

面内剪切非线性对复合材料层合板参数共振的影响

利用ＶｏｎＫａｒｍａｎ薄板大挠度理论和Ｈａｈｎ－Ｔｓａｉ本构方程研究了四边简支、一对边受压缩动载荷的（０／９０）。对称铺设正交各向异性矩形层合板的参数振动问题。     

Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process

This paper is focused on nonlinear dynamics of a shell-shaped workpiece during high speed milling. The shell-shaped workpiece is modeled as a double-curved cantilevered shell subjected to a cutting force with time delay effects. Equations of motion are derived by using the Hamilton principle based on the classical shell theory and von Karman strain-displacement relation. The resulting nonlinear partial differential equations are reduced to a two-degree-of-freedom nonlinear system by applying the Galerkin approach. The averaging method is used to obtain four-dimensional averaged equations for the case of foundational parametric resonance and 1:2 internal resonance. Using a numerical method, the dynamics of the cantilevered shell-shaped workpiece is studied under time-delay effects, parametric excitation, and forcing excitation. It is found that time-delay parameters have great impact on chaotic motion. With increasing amplitude of forcing and parametric excitations, the shell-shaped workpiece exhibits different dynamic behavior.     

OGY法实现混沌控制的参数辩识研究

给出了在系统方程未知情况下，采用OGY控制策略，通过混沌的部分测试信息辩识控制参数的方法，并对Duffing方程及强迫Brusselator振子方程的参数辩识及混沌控制进行了数值计算；讨论了使用这一方法实现控制时的一些关键问题，给出了实现控制的一些重要参数；发现了对Duffing方程使用同一参数辩识值，可将混沌控制到不同的一倍周期轨道及高倍周期轨道上的新现象。     

对称铺设正交各向异性层合板的亚谐参数共振

本文应用奇异性理论讨论了对称铺设正交各向异性层合矩形板的亚谐参数共振问题。主要内容是用Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ方法结合Ｚ２－对称等变的概念，使分叉方程转化为代数方程的研究，同时给出了参数平面上不同参数域中各种可能的分叉曲线。     

大挠度微曲矩形薄板的动力性质

本文导出了由位移表示的带初始挠度的粘弹性大挠度矩形薄板的运动方程。在简支边界条件下,应用分歧理论和Melnikov方法,研究在周期外力作用下系统的动力行为。给出了出现次谐分枝和马蹄态的条件。     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.     

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.     

Dynamic instability of truncated conical shells under periodic axial load

Based on the Dynamic version of Donnell type basic equations neglecting bending deformations before instability, the parametric instability of truncated conical shells subjected to periodic axial load is studied under four different boundary conditions. Applying Galerkin's method, the basic equations are reduced to a system of coupled Mathieu equations, from which the instability regions are determined by using Hsu's results. As a numerical example, instability regions for a completely clamped shell are determined for relatively wide range of frequencies. The effects of static axial load as well as damping force on the instability regions are also examined.