基于增量谐波平衡方法的Spar平台垂荡纵摇耦合内共振响应研究

针对经典的Spar平台垂荡纵摇耦合运动问题,为解决传统小参数摄动方法和时间步进分析方法的不足,提出将增量谐波平衡方法(Incremental Harmonic Balance Method,IHBM)应用于研究其内共振响应特性。根据Floquet稳定性分析理论,对周期解的稳定性和分叉特性进行分析;在此基础上,通过将该方法与增量弧长法相结合,实现了快速、连续获得Spar平台垂荡纵摇耦合周期运动响应的目的;将IHBM计算结果与时域模拟和多尺度法计算结果进行对比,验证了该方法的准确性和高效性,该方法能够准确预测当波浪激励力频率满足一定条件,系统发生内共振时引起的纵摇不稳定运动现象。对于垂荡纵摇耦合产生的概周期运动,该方法结合Floquet理论能准确预测其发生的参数区间,从而为该周期运动的分析提供良好的基础。     

Nonlinear analysis of an elastic cable under harmonic excitation

The nonlinear behavior of an elastic cable subjected to harmonic excitation is studied and solved. The method of multiple scales perturbation is applied to analyze the response of the nonlinear system near the simultaneous principle primary and internal resonance. The stability of the proposed analytic nonlinear solution near the simultaneous primary-internal resonance is studied and the stability condition is investigated. The effect of different parameters on the steady state responses of the vibrating system is studied and discussed using frequency response equations. The numerical solutions and chaotic response of the nonlinear system of the elastic cable for different parameters are also studied.     

分数导数黏弹性准饱和土中球空腔振动特性

在频率域内用解析方法研究分析了简谐轴对称荷载和流体压力作用下分数导数黏弹性准饱和中球空腔的稳态响应问题。将土骨架等效为具有分数阶导数本构关系的黏弹性体,基于Biot两相饱和介质模型,通过势函数推导求得了边界部分透水时分数导数粘弹性准饱和土中球空腔的位移、应力和孔压等的解析解。根据界面连续性条件,确定了待定系数的表达式。在此基础上,考察了准饱和土各参数对动力响应的影响,结果表明：轴对称荷载和流体压力两种情况时,相对渗透系数对动力响应的影响有较大的差异。分数导数本构模型更合理地描述了准饱和土中球空腔的振动特性。     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

基础激励下非线性碟簧减振系统的特性研究

详细研究了考虑静力载荷影响、具有弹性非线性和Coulmb摩擦阻尼非线性的磺簧元件的动力学特性。首先利用Fourier级数展开与平均法研究非线性碟簧系统的自由振动,讨论影响自由振动的振幅和频率的各种因素;然后利用Fourier级数展开与谐波平衡法研究基础激励下非线性簧系统的强迫振动,通过详细的参数研究考察非线性碟簧元件的减振性能特征;最后采用修正的稳定分析方法研究稳态响应的稳定性。     

叠层板在两项简谐激励作用下的组合共振分析

研究了在四边简支的边界条件下,正交各向异性矩形叠层板在两项横向简谐激励作用下的非线性组合共振及其稳定性问题。在给出了正交各向异性叠层板的振动微分方程的基础上,利用伽辽金法导出了相应的无量纲化达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解,得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件。通过数值算例,对几种复合材料薄板的共振特性进行了分析,分别给出了不同条件下系统运动的响应图、幅频图和动相平面图,讨论了不同参数对系统非线性动力学行为的影响。     

基于增量谐波平衡法的齿轮系统非线性动力学

利用增量谐波平衡法研究了一类考虑时变啮合刚度和间隙的直齿轮副的非线性动力学,建立了这类模型的解的统一形式。同时研究了系统的分岔特性以及阻尼比、外激励幅值对系统幅频曲线的影响。利用本文结果可以得到任意精度的近似解,从而可以更好地分析和控制齿轮系统的动力学。最后,指出了其它文献中未注意到的问题。     

滞后阻尼体系地震反应的中心差分虚初始条件法

针对滞后阻尼体系的直接积分法解发散问题,提出了中心差分虚初始条件法。该方法以实数初始条件为基础,建立相应的虚初始条件,以消除自由振动中的发散项,而使直接积分法的计算结果收敛到稳定解;并以有条件稳定的中心差分数值计算方法为基础,建立中心差分虚初始条件法的直接积分程序;对三个不同自振频率体系在三条地震波作用下进行中心差分虚初始条件法、解析解和频域解的数值计算。计算结果表明:频域分析结果仅包含稳态解,这将导致低自振频率体系在振动的初始阶段将产生显著误差;中心差分虚初始条件法对不同工况的计算结果都是稳定的,且收敛到包括瞬态反应的解析解。     

基于动力特性的混合结构地震响应复模态叠加法

混合结构由不同阻尼特性的材料组成,确定其阻尼矩阵存在困难。分块Rayleigh阻尼模型由于数学上的简易性,被广泛用于构建混合结构的阻尼矩阵,但分块Rayleigh阻尼模型的计算精度与参考频率的选择方法直接相关。针对参考频率的选择问题,依据结构动力响应的组成和特点,提出了一种确定Rayleigh阻尼系数的计算方法,进而实现基于分块Rayleigh阻尼模型的复模态叠加法。求解结构的瞬态反应时,根据结构前两阶振型的自振频率确定阻尼系数;求解结构的稳态反应时,选择结构的基频、与外激励频率接近的结构自振频率确定阻尼系数。依据地震波的频谱特性,提出了基于地震波卓越频率的分块Rayleigh阻尼模型,并结合地震加速度的分段线性假定,建立了混合结构的复模态叠加法。在此基础上,利用三角级数展开得到组成地震波的谐波频率,进一步提出了基于谐波频率的分块Rayleigh阻尼模型和对应的复模态叠加法。算例分析结果表明:所提方法误差更小,且克服了传统方法因振型选择不唯一导致的计算结果具有不确定性的问题;与基于地震波卓越频率的复模态叠加法相比,基于谐波频率的复模态叠加法计算量更大,但计算精度更高、适用范围更广。     

粘弹性复合材料夹芯板的稳态响应分析

芯材采用Kelvin粘弹性本构模型,推导了复合材料夹芯板的动力学方程,运用模态正交原理,以Navier完备解形式求解了四边简支正交对称铺层层合板的稳态响应,并给出了固有频率和结构损耗因子的解析解。通过固有频率的有限元解对比验证了数值计算的可靠性。分析了芯材剪切模量和芯材厚度对结构固有频率和损耗因子的影响。探讨了稳态响应的收敛性,并得到结构稳态响应振幅与频率的关系,分析了芯材损耗因子对结构稳态响应的影响。结果表明：芯材剪切模量存在最佳设计值;结构首阶模态特性主导结构的稳态动态响应。     

窄带激励下带有时滞反馈的非线性动力系统的响应

研究了窄带激励下带有时滞反馈的Duffing振子的主共振响应。用多尺度法分离了系统的快慢变量。分析了稳态响应的稳定性和分叉，研究了时滞、调谐参数、噪声带宽和幅值对系统的影响。证明了由于时滞的存在，系统将表现出复杂的动力学行为：时滞会导致分叉、时滞会影响跳跃区域等;发现噪声幅值会导致系统多解或分叉现象的出现，且随着噪声带宽的增大系统非零稳态响应从-极限环变为-扩散的极限环。最后，给出了数值模拟。     

Large amplitude vibration analysis of composite beams: Simple closed-form solutions

Large amplitude vibration analysis of laminated composite beam with axially immovable ends is investigated with symmetric and asymmetric layup orientations by using the Rayleigh–Ritz (R–R) method. The displacement fields used in the analytical formulation are coupled by using the homogeneous governing static axial equilibrium equation of the beam. Geometric nonlinearity of von-Karman type is considered which accounts for the membrane stretching action of the beam. The simple closed-form solutions are presented for the nonlinear harmonic radian frequency as function of central amplitude of the beam using the R–R method. The nonlinear harmonic radian frequency results obtained from the closed-form solutions of the R–R method in general show good agreement with the results obtained from simple iterative finite element formulation. Furthermore, the closed-form expressions are corrected for the harmonic motion assumption from the available literature results on the existence of quadratic and cubic nonlinearity. It is interesting to note that the composite beams can result in asymmetric frequency vs. amplitude curves depending upon the nature of direction of displacement in contrast to isotropic beams which exhibit cubic nonlinearity only and leads to symmetric frequency vs. amplitude curves with respect to sign of the amplitude.     

Dynamic Pareto fronts and optimal control of geometry in a problem of transient magnetic diffusion

The feasibility of dynamic multi-objective optimisation in computational electromagnetism is proved and a relevant benchmark of inverse magnetic diffusion is defined and solved. Accordingly, the optimal control of the geometry of a magnetic pole under step excitation has been proposed as a dynamic optimisation problem, characterised by two constrained objective functions that are both time- and field-dependent; the non-dominated solutions at steady state are to be determined. The benchmark has been solved numerically as a problem of transient magnetic diffusion, whereas the associated Pareto front has been identified by means of an enumerative search method in the time domain. In particular, the effect of a time-dependent energy constraint on the front at steady state has been determined. The theory of dynamic multi-objective optimisation in electromagnetism is discussed.     

Interior-Point Methods in l 1 Optimal Sparse Representation Algorithms for Harmonic Retrieval

Atomic decomposition is an alternative method for frequency detection in harmonic signals. This type of method produces very concentrated solutions with few nonzero components. It can be used as an alternative to traditional approaches, such as, principal-components frequency estimation methods. In this paper, we consider the basis pursuit principle to find the representation (frequency) coefficients of a harmonic signal by minimizing the l ₁ norm. For the l ₁ minimization, we compare two interior-point methods. A primal-dual method, which consists of the perturbed optimality conditions of the linear program, results in solutions that are more accurate and sparse than using a primal (affine scaling) method to solve the same linear program. We contrast the solutions obtained by the interior-point methods using the size of the given data and a bound for perfect recovery of the harmonic signals to establish the better performance of the primal-dual method. In addition, experimental results are shown.     

Vibration Performance,Stability and Energy Transfer of Wind Turbine Tower via Pd Controller

In this paper, we studied the vibration performance, energy transfer and stability of the offshore wind turbine tower system under mixed excitations. The method of multiple scales is utilized to calculate the approximate solutions of wind turbine system. The proportional-derivative controller was applied for reducing the oscillations of the controlled system. Adding the controller to single degree of freedom system equation is responsible for energy transfers in offshore wind turbine tower system. The steady state solution of stability at worst resonance cases is studied and examined. The offshore wind turbine system behavior was studied numerically at its different parameters values. Furthermore, the response and numerical results were obtained and discussed. The stability is also analyzed using technique of phase plane and equations of frequency response. In addition, the numerical results and comparison between analytical and numerical solutions were obtained with MAPLE and MATLAB algorithms.     

Winkler地基上四边自由矩形薄板的3次超谐波共振与奇异性

通过Galerkin方法,将Winkler地基上四边自由受横向简谐激励矩形薄板的控制微分方程转化为非线性振动方程。应用非线性振动的多尺度法,求得了系统满足3次超谐共振情况时的一次近似解以及对应的定常运动,并对其进行数值了计算。对3次超谐共振定常运动分岔响应方程进行了奇异性分析,得到了开折参数平面的转迁集和分岔图。揭示了一些新的动力学现象。     

Exact solutions of axial vibration problems of elastic bars

While exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to obtain exact solutions for free vibration and harmonic analyses for non‐trivial cases. This paper presents a study on new finite element formulation and algorithms for exact solutions of undamped axial vibration problems of elastic bars. Appropriate shape functions are constructed by using the homogeneous governing equations, and based on the new shape functions, a novel element is formulated. An iterative procedure is proposed for determining both the exact natural frequency values and the corresponding vibration mode shapes. Exact solutions can also be obtained for undamped harmonic response analyses by using the new element, as its stiffness and mass matrices are exact for a specified frequency. Illustrative examples are presented to demonstrate the effectiveness of the proposed element and algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

电阻电感非线性RLC电路弹簧耦合系统2次超谐共振

研究电阻和电感非线性RLC(Resistance-Inductance-Capacitance)电路弹簧耦合系统的非线性振动,应用拉格朗日—麦克斯韦方程,建立受简谐激励的具有电阻和电感非线性RLC电路弹簧耦合系统的数学模型。根据非线性振动的多尺度法,得到系统满足2次超谐共振条件的一次近似解以及对应的定常解。对其进行数值计算,分析系统参数对响应曲线的影响。增大激励电压和极板面积和电阻R1,响应曲线的振幅和共振区变大。增大极板间距、电感非线性系数、电阻R0和电阻R2,响应曲线的振幅和共振区变小。系统的固有频率随极板间距增大而增大,随极板面积和线性电感系数的增大而减小。     

复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。      

Winkler地基上四边自由矩形薄板的主共振与奇异性

通过Galerkin方法，将Winkler地基上四边自由受横向简谐激励矩形薄板的控制微分方程转化为非线性振动方程。应用非线性振动的多尺度法，求得了系统满足主共振情况时的一次近似解以及对应的定常运动，并对其进行数值计算，分析了激振力、调谐值、阻尼系数、非线性参数对系统的影响。对主共振定常运动分岔响应方程进行了奇异性分析，得到了开折参数平面的转迁集和分岔图。揭示了一些新的动力学现象。