Repelling Dynamics Near a Bykov Cycle

What set does an experimenter see while he simulating numerically the dynamics near a Bykov cycle? In this paper, we discuss the fate of typical trajectories near a Bykov cycle for a $C^1$ -vector field and we establish that despite the existence of shift dynamics (chaos) nearby, Lebesgue—almost all trajectories starting in a small neighbourhood of a Bykov cycle are repelled.     

Impact Dynamics in Milling of Thin-Walled Structures

The development of reliable high-speed spindles and motioncontrol systems has led to an increase in the industrial use ofhigh-speed milling. One of the primary applications of this newtechnology is the manufacture of thin-walled aluminum components foraircraft. The flexibility of the tools and workpieces, the high spindlefrequencies, and the inherent impact nonlinearities in the millingprocess can lead to complicated dynamic tool-workpieceinteractions. An experiment was constructed to study the vibrations ofa thin-walled part during milling. Time series, power spectra,autocorrelations, auto-bispectra, and phase portraits were examined.From this data, it is inferred that stiffness and damping nonlinearitiesdue to the intermittent cutting action have a pronounced effect on thedynamics of the workpiece. Delay space reconstructions and pointwisedimension calculations show that the associated motions arecharacterized by a fractal geometry. The auto-bispectra suggestquadratic phase coupling among the spectral peaks associated with thecutter frequency. A mechanics-based model with impact-nonlinearities wasdeveloped to explain the observed results. The predicted results agreewell with the experimental observations. The model predictions indicatethat aperiodic motions are possible over a large range ofcontrol-parameter values. These analytical and experimental results haveimplications for the prediction and control of vibrations in milling.     

Component-Mode Synthesis Method Variants in the Dynamics of Coupled Structures

The component-mode synthesis method is usually adopted in order to reduce the number of degrees-of-freedom of structures composed of two or more substructures without loosing the main physical characteristics of the whole structure. Many approaches of this method have been proposed in the literature. These approaches differ from each other for the boundary conditions which are imposed at the interface of the two substructures. In this paper four variants of interface boundary conditions are examined. For each set of conditions a suitable coordinate transformation, compatible with the conditions at the boundary degrees-of-freedom between the two substructures, is presented. Moreover, in the numerical applications a comparison between the four component-mode synthesis variants here proposed with respect to the same variants proposed in the literature is presented. The better accuracy of the proposed approach is shown.     

含对称间隙的摩擦振子非线性动力学分析

建立了两自由度含对称间隙的干摩擦碰撞振动系统的动力学模型,分析了系统运动中存在的滑动、黏着及碰撞,分别给出其判断方法和衔接准则,推导出各阶段系统的解析解,并采用数值迭代方法求解和分析了系统的复杂动力学行为,同时分析干摩擦对系统动力学性能的影响.结果表明,系统存在叉式分叉,系统由对称周期运动变为反对称周期运动,进而通过Hopf分叉或周期倍化分叉通向混沌.在参数变化范围较大的情况下,系统存在类型丰富的周期运动、拟周期运动以及混沌;系统存在对称运动、反对称运动对、黏滑碰撞运动以及由初始条件决定的共存吸引子.     

Stability of Spherically Symmetric Steady States in Galactic Dynamics¶against General Perturbations

Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a non-linear stability property of such steady states. In previous investigations by Y. Guo and G. Rein, stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this non-physical restriction.     

Dynamics of Sulfate Aerosol Formation in Engine Jets

The effect of SO₂, SO₃, HSO₃, and H₂SO₄ formation in the jet duct on the dynamics of sulfate H₂O/H₂SO₄ aerosol formation in the wake of a subsonic aircraft is analyzed numerically. It is shown that the presence, in addition to SO₂, of SO₃, HSO₃, and H₂SO₄ at the nozzle edge leads to an increase in the nucleation rate and intensification of the coagulation process. A significant quantity of large particles (of radius greater than 9 nm) is formed in the jet. This quantity depends on the sulfur content of the fuel.     

The Dynamics of Thin-Walled Structures Under Nonstationary Loads

Studies are reviewed in which nonstationary loads determining the dynamics of thin-walled structures are found and the initial concepts are formulated for the theory and methods of stress–strain analysis of shells with holes and different discrete reinforcements, as well as for coupled problems of hydro- and aeroelasticity. Special attention is drawn to the design and strength and dynamic analyses of thin-walled blasting chambers. The basic investigations conducted at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine are outlined     

Dynamics of Articulated Structures. Part I. Theory

ABSTRACT

A finite element based method is developed for geometrically nonlinear dynamic analysis of spatial articulated structures; i.e., structures in which kinematic connections permit large relative displacement between components that undergo small elastic deformation. Vibration and static correction modes are used to account for linear elastic deformation of components. Kinematic constraints between components are used to define boundary conditions for vibration analysis and loads for static correction mode analysis. Constraint equations between flexible bodies are derived in a systematic way and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A lumped mass finite element structural analysis formulation is used to generate deformation modes. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed in Part II of the paper.     

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.     

Dynamics of Miniature and High-Compliance Structures: Experimental Characterization and Modeling

Experimental Mechanics - The dynamic behavior of miniature and high-compliance structures is critical for their performance. However, their low stiffness and inertia bring significant challenges to...     

Wavelet Analysis of Structures: Statics, Dynamics and Damage Identification

Applications of the wavelet transform in solid mechanics are presented herein. The analysis problem is addressed first where the objective is to derive an adaptive wavelet-based method for the static and dynamic analysis of structures. This is done via a collocation scheme. The damage identification problem is investigated next making use of the space-localized properties of the wavelet transform: the regularity of the solution is detected by looking at the amplitude of the coefficients of the wavelet decomposition of the response. Open problems are finally outlined that will be the object of future work.     

Pattern Formation in Turing Systems on Domains with Exponentially Growing Structures

Turing reaction–diffusion systems have been used to model pattern formation in several areas of developmental biology. Previous biomathematical Turing system models employed static domains which failed to incorporate the growth that inherently occurs as an organism develops. To address this shortcoming, we incorporate an exponentially growing domain into a Turing system, allowing one to more realistically model biological pattern formation. This Turing system can generate patterns on an exponentially growing domain in any of the eleven coordinate systems in which the Helmholtz equation is separable, making the system incredibly flexible and giving one the capability to mathematically model pattern formation on a geometrically diverse group of domains. Linear stability analysis is employed to generate mathematical conditions which ensure such a system can generate patterns. We apply the exponentially growing Turing system to a prolate spheroidal domain and conduct numerical simulations to investigate the system’s pattern-generating behavior. We find that the addition of growth to a Turing system causes a significant change in the pattern-generating behavior of the system. While a static domain Turing system converges to a final pattern, an exponentially growing domain Turing system produces transient patterns that continually evolve and increase in complexity over time.     

Seventh Conference on Nonlinear Vibrations,Stability, and Dynamics of Structures

Nonlinear Dynamics -     

轴向运动二维传输结构动力学有限元建模与仿真

基于物理中面概念和经典薄板理论,应用有限元法研究了机械工程中的二维传输结构作轴向运动时的面外自由振动特性.根据实际工程结构特点及设计要点,考虑受双向预张应力作用的传输薄板结构模型,由哈密顿原理出发严格导出了结构的有限元动力学方程,得到了体现轴向传输结构特性的陀螺矩阵.该矩阵具有反对称结构,这与加权余量法所得的陀螺矩阵结构不同.采用3节点三角形单元离散求解域,且单元不受轴向运动影响,给出了单元密度对计算结果精度的影响.分析了传输结构预张应力和轴向速度与自由振动固有频率的关系;考察了不同结构的陀螺矩阵对数值结果的影响.将部分结果与ANSYS软件模拟对比,显示出良好的一致性,证明了本文方法的有效性.研究结果可为典型传输带等结构的振动控制提供参考,建模方法可为ANSYS等计算软件添加轴向运动结构新模块提供理论依据.     

Coherent Structures Formation During Wake-Boundary Layer Interaction on a LP Turbine Blade

Particle Image Velocimetry (PIV) measurements have been analyzed in order to characterize the dynamics of coherent structures (eddies and streaks) within the suction side boundary layer of a low pressure turbine cascade perturbed by impinging wakes. To this end, the instantaneous flow fields at low Reynolds number and elevated free-stream turbulence intensity level (simulating the real condition of the blade row within the engine) were investigated in two orthogonal planes (a blade-to-blade and a wall-parallel plane). Proper Orthogonal Decomposition (POD) has been employed to filter the instantaneous flow maps allowing a better visualization of the structures involved in the transition process of the boundary layer. For the unsteady case properly selected POD modes have been also used to sort the instantaneous PIV images in the wake passage period. This procedure allows computing phase-averaged data and visualizing structures size and intensity in the different parts of the boundary layer during the different wake passage phases. The contributions to the whole shear stress due to the largest spanwise oriented scales at the leading and trailing boundaries of the wake-jet structures and those associated with streaky structures observed in the bulk of the wake are discussed. Instantaneous images in the wall-parallel plane are filtered with POD and they allow us to further highlight the occurrence of low and high speed traveling streaks (Klebanoff mode). The periodic advection along the suction side of the high turbulent content regions carried by the wakes anticipates both formation and sinuous instability of the streaks inside the boundary layer as compared with the steady case. The dynamics driving the breakdown of the streaks and the consequent formation of nuclei with high wall-normal vorticity have been found to be almost the same in the steady and the unsteady cases. Auto-correlation of the instantaneous images are also presented in order to highlight analogies and differences in the size and spacing of streaks in the two cases. These results are also compared with the available literature concerning simplified geometries (i.e flat plate) operating under steady inflow.     

On the Averaging of Symmetric Positive-Definite Tensors

In this paper we present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by Cowin and Yang (J. of Elasticity 46 (1997) pp. 151–180.) for the case of the elasticity tensor of generalized Hooke's law.     

Symmetric failure of the halfspace with penny-shaped cracks in compression

The three-dimensional axisymmetric problem is investigated for a halfspace with penny-shaped crack parallel to the free surface. A uniform compression is applied parallel to the crack plane. The Griffith-Irwin theory is not applicable to this configuration of load and crack geometry since all the stress intensity factors are zero. Instead, a stability criterion will be invoked within the framework of the three-dimensional linearized stability theory. Reference can be made to previous works [5,6] involving compressible and incompressible elastic bodies. Use was made of an arbitrary form of the elastic potential for high subcritical deformations and for two variants of the theory of small subcritical deformations that involved equal and unequal roots of the characteristics equation [6]. It was found that consideration of the mutual influence of the subsurface crack and the free surface results in a considerable reduction of the theoretical strength limit. This was previously derived for infinite material with a crack. Examples of potentials with equal roots are discussed: the Bartenev-Khazanovich (incompressible bodies) and harmonic potential (compressible bodies).