Nonlinear vibrations of functionally graded doubly curved shallow shells

Nonlinear forced vibrations of FGM doubly curved shallow shells with a rectangular base are investigated. Donnell’s nonlinear shallow-shell theory is used and the shell is assumed to be simply supported with movable edges. The equations of motion are reduced using the Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Using the multiple scales method, primary and subharmonic resonance responses of FGM shells are fully discussed and the effect of volume fraction exponent on the internal resonance conditions, softening/hardening behavior and bifurcations of the shallow shell when the excitation frequency is (i) near the fundamental frequency and (ii) near two times the fundamental frequency is shown. Moreover, using a code based on arclength continuation method, a bifurcation analysis is carried out for a special case with two-to-one internal resonance between the first and second doubly symmetric modes with respect to the panel’s center (ω₁₃≈2ω₁₁). Bifurcation diagrams and Poincaré maps are obtained through direct time integration of the equations of motion and chaotic regions are shown by calculating Lyapunov exponents and Lyapunov dimension.     

Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly applied load is sufficiently large, the oscillation of the arch may reach a position at its primary unstable equilibrium path or secondary bifurcation unstable equilibrium path, leading the arch to buckle dynamically. This paper presents an analytical study of the nonlinear dynamic in-plane buckling of a shallow circular arch under a uniform radial load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. It is shown that under a suddenly applied uniform radial load, a shallow pinned–fixed arch has a unique possible dynamic buckling load, while shallow pinned–pinned and fixed–fixed arches may have two possible dynamic buckling loads: a lower dynamic buckling load and an upper dynamic buckling load. The dynamic buckling loads of a shallow arch under a suddenly applied uniform radial load with infinite duration are found to be lower than their static counterparts, and to increase with an increase of the arch included angle and slenderness. The effect of static preloading on the dynamic buckling of an arch is also investigated. It is found that the pre-applied static load decreases the dynamic buckling load of the arch, but increases the sum of the pre-applied load and the dynamic buckling load.     

负能模式在驱动漂移波非线性不稳定性中的作用(Ⅱ)——与正能模式交换,“回避交叉”和Hopf分岔

在正弦波驱动的非线性漂移波中,当驱动波频率和强度改变时,一个负能模式与另一个正能模式的复本征值可能出现交叉和“回避交叉”,在出现“回避交叉”的同时,这两个模式的地位发生了某种交换。正负能量两个模式之间的这种非线性共振在弱耗散下可能引起Hopf分岔。     

Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling

A two-dimensional nonlinear Schrödinger lattice with nonlinear coupling, modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear Kerr material, is studied. We find that despite a vanishing energy difference (Peierls-Nabarro barrier) of fundamental stationary modes the mobility of localized excitations is very poor. This is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes. At these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points. The control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power-coupling devices. In certain parameter regimes, the direction of a stable propagating plane-wave current is shown to be continuously tunable by amplitude variation (with fixed phase gradient). More exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices, and stationary complex modes with nontrivial phase relations are also briefly discussed. Regimes of dynamical linear stability are found for all these types of solutions.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

有内共振时的扬声器分谐波和混沌

研究了当轴对称模态由驱动力共振激发,并且轴对称模态和非轴对称模态存在2:1内共振时的扬声器辐射体薄壳的分谐波和昆沌。采用多尺度法分析了非线性模态方程的稳态解及其稳定性,由此进一步确定了驱动频率和驱动力平面上的分岔集。给出了所考虑情形下扬声器分谐波的阈值电压公式,该阈值电压低于无内共振时的阈值电压。除出现非轴对称模态的1/2分谐波振动外,2个模态的振幅经Hopf分岔后作极限环运动,并经倍周期分岔进入混沌运动。混沌出现是由于2个模态间能量的强烈交换。理论结果和实验结果基本吻合,该一致性表明了所建扬声器非线性薄壳模型的正确性。      

Multiple internal resonances and non-planar dynamics of shallow suspended cables to the harmonic excitations

In the present study, the nonlinear response of a shallow suspended cable with multiple internal resonances to the primary resonance excitation is investigated. The method of multiple scales is applied directly to the nonlinear equations of motion and associated boundary conditions to obtain the modulation equations and approximate solutions of the cable. Frequency–response curves and force–response curves are used to study the equilibrium solution and its stability. The effects of the excitation amplitude on the frequency–response curves of the cable are also analyzed. Moreover, the chaotic dynamics of the shallow suspended cable is investigated by means of numerical simulations.     

耦合相对转动非线性动力系统的稳定性与近似解

研究了一类含三次非线性耦合项的相对转动非线性动力系统的动力学行为. 建立了具有非线性弹性力、广义摩阻力耦合项的系统动力学方程. 运用多尺度法求解谐波激励下耦合非自治系统的近似解,通过讨论系统的主共振和内共振特性,分析了耦合项对系统响应的影响. 应用奇异性理论研究了主振稳态响应分岔方程的稳定性,得到了系统的转迁集和分岔曲线的拓扑结构. 关键词： 相对转动 非线性耦合动力系统 奇异性理论 稳定性     

Large-amplitude nonlinear vibrations of a Mooney–Rivlin rectangular membrane

An analysis of the linear and nonlinear vibration response and stability of a pre-stretched hyperelastic rectangular membrane under harmonic lateral pressure and finite initial deformations is presented in this paper. Geometric nonlinearity due to finite deformations and material nonlinearity associated with the hyperelastic constitutive law are taken into account. The membrane is assumed to be made of an isotropic, homogeneous, and incompressible Mooney–Rivlin material. The results for a neo-Hookean material are obtained as a particular case and a comparison of these two constitutive models is carried out. First, the exact solution of the membrane under a biaxial stretch is obtained, being this initial stress state responsible for the membrane stiffness. The equations of motion of the pre-stretched membrane are then derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are analytically obtained for both materials. The natural modes are then used to approximate the nonlinear deformation field using the Galerkin method. A detailed parametric analysis shows the strong influence of the stretching ratios and material parameters on the linear and nonlinear oscillations of the membrane. Frequency–amplitude relations, resonance curves, and bifurcation diagrams, are used to illustrate the nonlinear dynamics of the membrane. The present results are compared favorably with the results evaluated for the same membrane using a nonlinear finite element formulation.     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

Bifurcation and resonance in a fractional Mathieu-Duffing oscillator

The bifurcation and resonance phenomena are investigated in a fractional Mathieu-Duffing oscillator which contains a fast parametric excitation and a slow external excitation. We extend the method of direct partition of motions to evaluate the response for the parametrically excited system. Besides, we propose a numerical method to simulate different types of local bifurcation of the equilibria. For the nonlinear dynamical behaviors of the considered system, the linear stiffness coefficient is a key factor which influences the resonance phenomenon directly. Moreover, the fractional-order damping brings some new results that are different from the corresponding results in the ordinary Mathieu-Duffing oscillator. Especially, the resonance pattern, the resonance frequency and the resonance magnitude depend on the value of the fractional-order closely.     

Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory

In this paper, controlling chaos when chaotic ferroresonant oscillations occur in a voltage transformer with nonlinear core loss model is performed. The effect of a parallel metal oxide surge arrester on the ferroresonance oscillations of voltage transformers is studied. The metal oxide arrester(MOA) is found to be effective in reducing ferroresonance chaotic oscillations. Also the multiple scales method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes sub-harmonic, quasi-periodic, and also chaotic oscillations. In this paper, the chaotic behavior and various ferroresonant oscillation modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as period doubling bifurcation(PDB), saddle node bifurcation(SNB), Hopf bifurcation(HB), and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via the multiple scales method to obtain Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system.     

Nonlinear dynamics of a pipe conveying pulsating fluid with combination,principal parametric and internal resonances

Nonlinear dynamics of a hinged–hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated. The system has geometric cubic nonlinearity due to stretching effect out of immovable support conditions at both ends. The pipe conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. For appropriate choice of system parameters, the natural frequency of the second mode is approximately three times that of the first mode for a range of mean flow velocity, activating a three-to-one internal resonance. The analysis is carried out using the method of multiple scales by directly attacking the governing nonlinear integro-partial-differential equations and the associated boundary conditions. The set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for combination parametric resonance and principal parametric resonance. Stability, bifurcation and response behavior of the pipe are investigated. The amplitude and frequency detuning of the harmonic velocity perturbation are taken as the control parameters. The system exhibits response in the directly excited and indirectly excited modes due to modal interaction. Dynamic response of the system is presented in the form of phase plane trajectories, Poincare maps and time histories. A wide array of dynamical behavior is observed illustrating the influence of internal resonance.     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.     

Duffing系统的主-超谐联合共振

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到稳态周期解的稳定性条件,并分析了非线性刚度对稳态周期解的幅值和稳定性的影响.此外,由于近似解只能描述周期运动,不足以描述系统的全局特性,因而应用Melnikov方法对系统进行全局分析,得到系统进入Smale马蹄意义下混沌的条件,依据该条件以及主-超谐联合共振的条件选取一组参数进行数值仿真.分岔图和最大Lyapunov指数显示出两个临界值:当激励幅值通过第一个临界值时,异宿轨道破裂,混沌吸引子突然出现,系统以激变方式进入混沌;激励幅值通过第二个临界值时,系统在混沌态下再次发生激变,进入另一种混沌态.利用Melnikov方法考察了第一个临界值在多种频率组合下的变化趋势,并用数值仿真验证了解析结果的正确性.     

二维抛物线离散映射的动力学研究

由两个一维抛物线离散映射作推广并非线性耦合,实现了一个新的二维抛物线离散映射.利用不动点稳定性分析和映射分岔分析,研究了所提出的二维离散映射的复杂动力学行为及其吸引子的演变过程,阐述了它所特有的共存分岔模式和快慢周期振荡效应等动力学特性.研究结果表明:二维抛物线离散映射具有动力学特性调节和动态幅度调节的两个功能不同的控制参数,存在Hopf分岔、分岔模式共存、锁频和周期振荡快慢效应等非线性物理现象.并基于微控制器实现的数字电路验证了相应的理论分析和数值仿真结果. 关键词： 二维离散映射 分岔 吸引子 参数     

Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions

Large-amplitude (geometrically nonlinear) forced vibrations of circular cylindrical shells with different boundary conditions are investigated. The Sanders-Koiter nonlinear shell theory, which includes in-plane inertia, is used to calculate the elastic strain energy. The shell displacements (longitudinal, circumferential and radial) are expanded by means of a double mixed series: harmonic functions for the circumferential variable and three different formulations for the longitudinal variable; these three different formulations are: (a) Chebyshev orthogonal polynomials, (b) power polynomials, and (c) trigonometric functions. The same formulation is applied to study different boundary conditions; results are presented for simply supported and clamped shells. The analysis is performed in two steps: first a liner analysis is performed to identify natural modes, which are then used in the nonlinear analysis as generalized coordinates. The Lagrangian approach is applied to obtain a system of nonlinear ordinary differential equations. Different expansions involving from 14 to 34 generalized coordinates, associated with natural modes of both simply supported and clamped-clamped shells, are used to study the convergence of the solution. The nonlinear equations of motion are studied by using arclength continuation method and bifurcation analysis. Numerical responses obtained in the spectral neighborhood of the lowest natural frequency are compared with results available in literature.     

A time delay control for a nonlinear dynamic beam under moving load

The bifurcation resulted from moving force may lead to instability for the system. Based on time delay feedback controller, a nonlinear beam under moving load is discussed in the case of the primary resonance and the 1/3 subharmonic resonance. The bifurcation may be eliminated or the bifurcation point's position may be changed. The perturbation method is used to obtain the bifurcation equation of the nonlinear dynamic system. The result indicates time delay feedback controller may work well on this system, but the selection of a proper time delay and its coefficient may depend on the engineering condition. This paper presents some theoretical results.     

有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质,发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为,适当选择这些参数可以改变系统的分岔响应.同时,还讨论了非零解的稳定性,得到了非零解稳定的充要条件,而且发现在随机激励的带宽较小时,系统的多解现象仍然存在,分岔和跳跃现象仍会发生,数值模拟验证了理论推导的有效性. 关键词： 随机Mathieu-Duffing系统 多尺度 稳定性 分岔