小参数法在悬垂索共振中的应用

当面外横向振动和面内横向振动频率的比接近1:2时,悬索会出现面内和面外耦合共振现象。为了研究悬索这种复杂独特的非线性特性,利用多尺度法对谐波激励的悬索动力学方程进行求解,得到对应于不同阶小量的偏微分方程组,其中二阶小量偏微分方程中的久期项不为0;采用提出的小参数法可以得到由久期项引起的悬索振动形态,解决久期项频率与系统频率相同但不能直接求解的问题;为了证明小参数法的准确性,采用Galerkin方法离散悬垂索的运动方程,然后利用多尺度法求解离散的运动方程,得到采用基函数描述的由久期项引起的连续系统的振动形态,与小参数法结论一致。     

悬索在考虑1:3内共振情况下的动力学行为

研究了悬索在受到外激励作用下考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.     

悬索主共振响应的多尺度解与同伦分析解

利用哈密顿变分原理,引入拟静态假设,建立了悬索面内非线性运动方程,并采用Galerkin方法对其进行离散。接着运用多尺度法和同伦分析法得到了悬索前两阶模态主共振响应的近似解。为验证这两种分析方法的适用性,同时采用龙格-库塔法对方程直接进行了数值积分。数值计算结果表明,随着悬索垂跨比以及振幅的增加,由多尺度法与同伦分析法得到的幅频响应曲线存在明显的定性与定量的差别,而同伦分析法结果与数值法的结果更加接近。最后比较了两种分析方法得到的位移场与索力时程响应曲线。     

悬索在外激励作用下的1∶3内共振分析I∶离散法

研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程.     

一种单元谐波平衡法

基于有限元离散,对于工程中的非线性响应问题,提出一种单元谐波平衡法．与常规的谐波平衡法不同,本文将谐波平衡方程建立在有限元素上,从而兼顾了有限元素法和常规谐波平衡法两大优势．有限元技术的应用能使得求解问题的范围扩大到复杂工程结构,而谐波平衡概念的使用将使得含有复杂变形和复杂本构关系的动力学响应问题得到有效解决．所提方法能适用于工程结构中具有复杂非线性关系的动力学响应问题．由于谐波平衡法的实施依赖于谐波系数方程及其切线刚度矩阵的解析推导,尽管已经局限到有限元素上,但对于较为复杂一些的本构关系,推导仍非易事．为解决这些问题,放弃通常对于变形梯度和应变张量所作的向量假设,而是从连续介质力学中基本的几何关系入手,提出一种矩阵分解形式．通过利用张量的内蕴导数定义以及关于迹函数的有关性质,给出应力增量的一种新的表现形式．当它与变形梯度的矩阵分解相结合时,使得切线刚度矩阵的导出变得十分简单,而且所得计算形式也比通常紧凑和方便许多．     

悬索在外激励作用下的1∶3内共振分析Ⅰ∶离散法

研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应。对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在。首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程。     

非线性转子系统周期响应的分析

基于时域的时间有限元法,将描述转子系统动力学特征的非线性微分方程组离散成一组非线性代数方程,然后应用吴消去法的特征列思维对所得到的非线性代数方程组进行降维求解,进而得到待求节点位移响应的解形式,并据此对一具有非线性支撑的柔性Jeffcott转子模型响应的性质进行了分析。     

悬索的多重内共振研究

本文对谐波激励的悬索的非线性响应进行了研究,同时考虑了如下问题(1):面内第三阶对称模态的主共振:(2):面内第一阶、第三阶对称模态和面外第五阶模态之间的内共振.本方首先针对考虑大变形的悬索动力学方程,由线性理论求得各阶频率,考察可能出现的内共振.然后利用直接法对悬索的运动学方程和边界条件进行非线性求解.由多尺度法得到系统的平均方程和悬索响应的二阶近似解.随后利用Newton-Raphson 方法和弧长法对特定张拉索进行数值仿真计算,得到面内第一阶对称模态、面内第三阶对称模态和面外第五阶模态的稳态解,并分析了解的稳定性.绘制幅频响应曲线,发现了关于悬索响应的多种分叉现象,并且对各种分叉现象周期解、混沌解进行了讨论.     

一种求解局部非线性结构稳态响应的方法

为了降低求解局部非线性结构稳态响应的计算量,基于子结构和阻抗缩聚提出了一种用于求解局部非线性结构稳态响应的计算方法.将局部非线性结构分解为线性子结构和非线性子结构,利用谐波平衡构造各个子结构的阻抗方程,对线性子结构进行缩聚,将局部非线性动力学方程转化为求解一组非线性代数方程组问题,通过迭代求解非线性代数方程组,求解系统的稳态响应.     

风作用下覆冰悬索的非平面非线性动力学

主要研究侧向风载荷作用下小垂度覆冰悬索的非线性非平面运动的复杂动力学.根据分析力学、弹性力学和空气动力学理论,建立覆冰悬索3个自由度非线性振动的偏微分运动方程,并对其进行无量纲化,运用Galerkin方法对偏微分运动方程进行离散得到3个自由度的常微分方程,再利用多尺度法得到面内主共振2:1内共振的平均方程.利用数值方法研究悬索的非线性运动,结果表明系统呈现周期、多倍周期、概周期和混沌运动的规律.     

Comparison of several models for multi-size bubbly flows on an adiabatic experiment

This paper deals with the modelling and numerical simulation of isothermal bubbly flows with multi-size bubbles. The study of isothermal bubbly flows without phase change is a first step towards the more general study of boiling bubbly flows. Here, we are interested in taking into account the features of such isothermal flow associated to the multiple sizes of the different bubbles simultaneously present inside the flow. With this aim, several approaches have been developed. In this paper, two of these approaches are described and their results are compared to experimental data, as well as to those of an older approach assuming a single average size of bubbles. These two approaches are (i) the moment density approach for which two different expressions for the bubble diameter distribution function are proposed and (ii) the multi-field approach. All the models are implemented into the NEPTUNE_CFD code and are compared to a test performed on the MTLOOP facility. These comparisons show their respective merits and shortcomings in their available state of development.     

Aeroelastic coupling of geometrically nonlinear structures and linear unsteady aerodynamics: Two formulations

Two different time domain formulations of integrating commonly used frequency-domain unsteady aerodynamic models based on a modal approach with full order finite element models for structures with geometric nonlinearities are presented. Both approaches are tailored to flight vehicle configurations where geometric stiffness effects are important but where deformations are moderate, flow is attached, and linear unsteady aerodynamic modeling is adequate, such as low aspect ratio wings or joined-wing and strut-braced wings at small to moderate angles of attack. Results obtained using the two approaches are compared using both planar and non-planar wing configurations. Sub-critical and post-flutter speeds are considered. It is demonstrated that the two methods lead to the same steady solution for the sub-critical case after the transients subside. It is also shown that the two methods predict the amplitude and frequency of limit cycle oscillation (when present) with the same accuracy.     

Continuum analytical modelling of thermal creep

A thermal creep process is studied in quite wide rectangular micro channels, sufficiently wide so that it is possible to consider this configuration as two parallel plates. The inlet and outlet reservoirs are maintained at the same constant pressure. A constant temperature gradient exists along the walls of the channel joining the two tanks. Thus a gas flow is induced and thermally sustained until steady conditions are reached. A complete analytical solution is derived in slip regime, yielding all the flow parameters, for Knudsen numbers smaller than 0.25. The analytical results are in good agreement with the numerical “exact” solution of the continuum equation system. Furthermore our continuum approach data are compared to those deduced from approaches based on Boltzmann equation model treatments: these various methods lead generally to a satisfactory agreement between their respective mean parameters. Nevertheless significant differences appear on the transversal velocity profiles and are further discussed.     

焦散线法求解裂纹应力强度因子的两种改进方法

为提高焦散线法求解应力强度因子的精度，提出了两种基于多点非线性最小二乘法的改进方法。文中推导了控制方程，编制了程序，阐述了实验技术。作为应用实例，对三种典型裂纹试件作了测试与处理。结果表明，用本文方法处理的结果比文［１］经典法的结果有很大改进。作者还对图像处理用于焦散线法应力强度因子的提取作了初步探讨［５     

Recent advances and comparisons between harmonic balance and Volterra-based nonlinear frequency response analysis methods

Harmonic balance and Volterra-based analysis methods are well known, but the capabilities of these methods have been limited by significant issues of complexity which either constrain their application to relatively simple cases, or limit the accuracy of analysis in more complex cases. This study briefly summarizes recent results which effectively extend the capabilities of both harmonic balance and Volterra-based analysis by making complex analyses much more feasible. The new capabilities and performance of the two approaches are then evaluated and compared using benchmark case studies of a Duffing oscillator and a nonlinear automotive damper. The results offer new insights and lead to different conclusions on the relative merits of harmonic balance versus Volterra-based analysis relative to prior studies and similar benchmark analyses.     

Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets

Finite element (FE) simulations of the simple shear test were conducted for 1050-O and 6022-T4 aluminum alloy sheet samples. Simulations were conducted with two different constitutive equations to account for plastic anisotropy: Either a recently proposed anisotropic yield function combined with an isotropic strain hardening law or a crystal plasticity model. The FE computed shear stress–shear strain curves were compared to the experimental curves measured for the two materials in previous works. Both phenomenological and polycrystal approaches led to results consistent with the experiments. These comparisons lead to a discussion concerning the assessment of anisotropic hardening in the simple shear test.     

From dynamic modulus via different relaxation spectra to relaxation and creep functions

The main goal of the paper is to compare predictive power of relaxation spectra found by different methods of calculations. The experimental data were obtained for a new family of propylene random copolymers with 1-pentene as a comonomer. The results of measurements include flow curves, viscoelastic properties, creep curves and rubbery elasticity of copolymer melts. Different relaxation spectra were calculated using independent methods based on different ideas. It lead to various distributions of relaxation times and their “weights”. However, all of them correctly describe the frequency dependencies of dynamic modulus. Besides, calculated spectra were used for finding integral characteristics of viscoelastic behaviour of a material (Newtonian viscosity, the normal stress coefficient, steady-state compliance). In this sense all approaches are equivalent, though it appears impossible to estimate instantaneous modulus. The most crucial arguments in estimating the results of different approaches is calculating the other viscoelastic function and predicting behaviour of a material in various deformation modes. It is the relaxation and creep functions. The results of relaxation curve calculations show that all methods used give rather similar results in the central part of the curves, but the relaxation curves begin to diverge when approaching the high-time (low-frequency) boundary of the relaxation curves. The distributions of retardation times calculated through different approaches also appear very different. Meanwhile, predictions of the creep curves based on these different retardation spectra are rather close to each other and coincide with the experimental points in the wide time range. Relatively slight divergences are observed close to the upper boundary of the experimental window. All these results support the conclusion about a rather free choice of the relaxation time spectrum in fitting experimental data and predicting viscoelastic behaviour of a material in different deformation modes. Received: 15 March 2000 Accepted: 18 September 2000     

Double-mode modeling of chaotic and bifurcation dynamics for a simply supported rectangular plate in large deflection

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate is first derived and simplified to a set of two ordinary differential equations by the Galerkin method. Several different features including Fourier spectra, state-space plot, Poinca?e map and bifurcation diagram are then numerically computed by using a double-mode approach. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of a rectangular plate possesses many bifurcation points, two different chaotic motions and some jump phenomena under various lateral loading. The results of numerical simulation indicate that the computed bifurcation points can lead to either a transcritical bifurcation or a pitchfork bifurcation for the motion of a large deflection rectangular plate. Meanwhile, the points of pitchfork bifurcation can gradually lead to chaotic motion in some specific loading conditions. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of a large deflection plate.     

Comparison of kinetic and continuum approaches for simulation of shock wave/boundary layer interaction

The present paper, which is a collaboration between three different research groups, analyzes the efficiency of various numerical approaches to describe the complex problem of shock wave/boundary layer interaction. Computations were carried out based on a kinetic approach (Direct Simulation Monte Carlo method) and on two continuum approaches (Navier-Stokes equations and quasigasdynamic equations), which are validated by comparison with experimental results obtained in the R5Ch blowdown Hypersonic Wind Tunnel in ONERA. The influence of the slip boundary conditions for two continuum approaches are also studied. The results obtained by all models display the good prediction of the main structure of the flow and the levels of the flux coefficients are very close to those measured. The implementation of the slip boundary condition for the continuum approaches improves the agreement with the experimental data. Received 12 July 2001 / Accepted 24 May 2002 /Published online 4 December 2002 Correspondence to: D. Zeitoun (e-mail: David.Zeitoun@polytech.univ-mrs.fr) An abridged version of this paper was presented at the 23rd Int. Symposium on Shock Waves at Fort Worth, Texas, from July 22 to 27, 2001     

Modelling of thin piezoelectric layers on plates

The derivation of plate equations for a plate consisting of two layers, one anisotropic elastic and one piezoelectric, is considered. Power series expansions in the thickness coordinate for the displacement components and the electric potential lead to recursion relations among the expansion functions. Using these in the boundary and interface conditions, a set of equations are obtained for some of the lowest-order expansion functions. This set is reduced to six equations corresponding to the symmetric (in-plane) and antisymmetric (bending) motions of the elastic layer. These equations are given to linear (for the symmetric equations) or quadratic (for the antisymmetric equations) order in the thickness. It is noted that it is, in principle, possible to go to any order, and that it is believed that the corresponding equations are asymptotically correct. A few numerical results for guided waves along the plate and a 1D actuator case illustrate the accuracy.