一种求解非线性振动系统渐近解的新方法——计算向量场Normal Form系数的简单方法

利用非线整变换,本文推导出了一种只需通过简单代数运算即可算出Hopf分叉Normal Form系数的简单方法,用这种方法求解非线性振动问题,只需把原方程变换成本文讨论的典则形式的一阶微分方程组,然后进行简单的代数运算即可得到原非线性振动方程的解,这种方法简单方便容易掌握。     

一种求解非线性振动系统渐近解的新方法——计算向量场Normal Form系数的简单方法

利用非线整变换,本文推导出了一种只需通过简单代数运算即可算出Hopf分叉Normal Form系数的简单方法,用这种方法求解非线性振动问题,只需把原方程变换成本文讨论的典则形式的一阶微分方程组,然后进行简单的代数运算即可得到原非线性振动方程的解,这种方法简单方便容易掌握。     

机电耦联系统余维3动态分岔研究

以r_{sl}, r_{f}以及x_{c}为分岔参数，对具有串补电容的单 机无穷大电力系统的失稳振荡问题，运用动态分岔理论进行了研究. 对系统同时出现有3对 纯虚根特征值的一类多参数高余维分岔情况，运用中心流行方法降维后得到约化方程，对此 强非线性约化方程的求解难点，运用多参数稳定性理论、谐波平衡法、归一化技术和Normal Form方法，得到了系统的解析解. 由分析得知，系统会出现3种Hopf分岔情况、二维环面 情况，以及三维环面分岔解，甚至会出现四维环面，或者更高维的环面分岔. 详细讨论 了系统各种分岔解的稳定性条件和稳定区域，并作了详细的数值分析加以验证.     

实现电子闭环控制的非线性动力学系统的周期解

本文对具有自动频率跟踪和反馈稳幅功能的电磁振动机械进行了研究.对实现电子闭环控制的一个半自由度非线性动力学系统(见图1)建立了一个三维机、电振动微分方程纽。应用中心流形定理和Normal Form定理对这类具有自激振动性质的方程组求得第一次近似解.又以GZJ—4型机为例,计算了定常解的稳定域,并给出定常解同电磁铁激磁电压之间的关系.对设计和调试这类产品有指导意义.     

非半单分叉的Normal Form计算

在文［１］基础上，提出一种仅知道派生线性系统零实部特征值时求解非线性系统非半单分叉ＮｏｒｍａｌＦｏｒｍ的方法．通过适当的分类，将要求解的线性代数方程组分为若干相互独立的方程组．将所求系数向量按字典序列排列后，各独立方程组的系数矩阵是上三角矩阵．在非共振情形，各系数向量可按反字典序列递推求出．在共振情形，根据文中的二个定理，巧妙地由一简单的常数矩阵的最大秩子矩阵，定位其系数矩阵的满秩子矩阵，解决了这类方程组的降维简化．通过消元法，把简化后的方程化成类似于半单分叉ＮｏｒｍａｌＦｏｒｍ求解过程中方程的形式，其解法也类似．该方法非常易于在计算机代数软件平台上程序化．     

1／2亚谐共振──主参数共振情况下非线性振动系统的余维2退化分叉

本文应用Normal Form理论和退化向量场的普适开折理论研究了参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉,用Melnikov方法讨论了全局分叉的存在性.     

非线性模态的分类和新的求解方法

引入不可分偶数维不变流形的概念来定义非线性模态．在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类．这种分类可能是新的构筑非线性模态理论的框架．用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息．不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统；不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态．从掌握的资料看，以前的方法还不能解决上述所有问题     

Duffing 系统的主-亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.      

Duffing系统的主--亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.     

加性非平稳激励下结构速度响应的FPK方程降维

结构在随机激励下的非线性响应分析是具有高度挑战性的困难问题. 对于白噪声或过滤白噪声激励,求解FPK方程将获得结构响应 的精确解. 遗憾的是,对于非线性多自由度系统,FPK方程难以直接求解. 事实上,其数值解法严重受限于方程维度,而解析求解 则仅适用于少数特定的系统,且多是稳态解. 因此,将FPK方程进行降维,是求解高维随机动力响应分析问题的重要途径. 本文针 对幅值调制的加性白噪声激励下多自由度非线性结构的非平稳随机响应分析问题,将联合概率密度函数满足的高维FPK方程进行降 维. 针对结构速度响应概率密度函数求解,通过引入等价漂移系数,原FPK方程可转化为一维FPK型方程. 建议了构造等价漂移系数 的条件均值函数方法. 进而,采用路径积分方法求解降维FPK型方程,得到速度概率密度函数的数值解答. 结合单自由度Rayleigh 振子、十层线性剪切型框架和非线性剪切型框架结构在幅值调制的加性白噪声激励下的非平稳速度响应求解,讨论了本文方法的精 度和效率,验证了其有效性.      

损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先基于哈密顿变分原理，引入与损伤程度、范围和位置相关的三个无量纲参数，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．     

Resonant dynamics in strongly nonlinear systems

This work considers the effect of resonances in systems in which the two resonant frequencies are allowed to slowly change, depending on the state of the system. A strongly nonlinear system is introduced that allows for exact solutions. This system is then coupled to a second component, and through the method of averaging, a reduced-order model is developed that approximates the dynamical behavior near a 1:1 resonance between the two components. The resulting reduced system is studied using bifurcation theory and Melnikov analysis to obtain predictions of the near-resonant dynamics. Finally, these predictions are compared to numerical simulations of the original equations. Two main points appear: (i) for nonlinear systems, the period–amplitude dependence plays an important role in the evolution of the system, and (ii) the coordinates identified within the reduced system allow for the qualitative structure of the original equations to appear.     

Internal resonance of pipes conveying fluid in the supercritical regime

This paper treats nonlinear vibration of pipes conveying fluid in the supercritical regime. If the flow speed is larger than the critical value, the straight equilibrium configuration becomes unstable and bifurcates into two possible curved equilibrium configurations. The paper focuses on the nonlinear vibration around each bifurcated equilibrium. The disturbance equation is derived from the governing equation, a nonlinear integro-partial-differential equation, via a coordinate transform. The Galerkin method is applied to truncate the disturbance equation into a two-degree-of-freedom gyroscopic systems with weak nonlinear perturbations. The internal resonance may occur under the certain condition of the supercritical flow speed for the suitable ratio of mass per unit length of pipe and that of fluid. The method of multiple scales is applied to obtain the relationship between the amplitudes in the two resonant modes. The time histories predicted by the analytical method are compared with the numerical ones and the comparisons validate the analytical results when the nonlinear terms are small.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

This paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second-order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω ₁, ω ₂, and ω ₃ such that ω ₃?2ω ₂?4ω ₁, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω ₃?ω ₂?2ω ₁. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics and favors the presence of unstable periodic orbits.     

Combination,principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.     

Nonlinear Dynamics of a Beam on Elastic Foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

Nonlinear energy harvesting with dual resonant zones based on rotating system

An electromagnetic nonlinear energy harvester(NEH)based on a rotating system is proposed,of which the host system rotates at a constant speed and vibrates harmonically in the vertical direction.This kind of device exhibits several resonant phenomena due to the combinations of the rotating and the vibration frequencies of the host system as well as the cubic nonlinearity of the NEH.The governing equation of motion for the NEH is derived,and the dynamic responses and output power are investigated with the multiple scale method under the 1:1 primary and 2:1 superharmonic resonant conditions.The effects of system parameters including the nondimensional external frequency,the rotating speed,and the nonlinear stiffness on the responses of free vibration for the system are studied.The results of the primary resonance show that the responses exhibit not only the resonant characteristics but also the nonlinear dynamic characteristics such as the saddle-node(SN)bifurcation.The coexistence of multiple solutions and the varying trends of responses are verified with the direct numerical simulation.Moreover,the effects of system parameters on the average output power are investigated.The results of the analyses on the two resonant conditions indicate that the large power can be harvested in two resonant frequency bands.The effect of resonance on the output power is dominant for the 2:1 superharmonic resonance.Moreover,the results also show that introducing the nonlinearity can increase the value of the output power in large frequency bands and induce the occurence of new frequency bands to harvest the large power.The efficiency of the harvested power could be improved by the combined effects of the resonance as well as the nonlinearity of the NEH device.Suitable parameter conditions could help optimize the power harvesting in design.