An analytic approximate approach for free oscillations of self-excited systems

A new analytic approximate technique for non-linear problems, namely the homotopy analysis method, is employed to propose an approach for free oscillations of self-excited systems. Different from perturbation methods on this topic, this approach does not depend upon any small/large parameters at all and therefore is valid for free oscillations of all self-excited systems. Besides, unlike other analytic techniques, this approach provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. Two examples are employed to illustrate the validity and flexibility of this approach.     

Frequency lock-in phenomenon for oscillating airfoils in buffeting flows

Navier-Stokes based computer simulations are conducted to determine the aerodynamic flow field response that is observed for a NACA0012 airfoil that undergoes prescribed harmonic oscillation in transonic buffeting flows, and also in pre-buffet flow conditions. Shock buffet is the term for the self-sustained shock oscillations that are observed for certain combinations of Mach number and steady mean flow angle of attack even in the absence of structural motion. The shock buffet frequencies are typically on the order of the elastic structural frequencies, and therefore may be a contributor to transonic aeroelastic response phenomena, including limit-cycle oscillations. Numerical simulations indicate that the pre-shock-buffet flow natural frequency increases with mean angle of attack, while the flow damping decreases and approaches zero at the onset of buffet. Airfoil harmonic heave motions are prescribed to study the interaction between the flow fields induced by the shock buffet and airfoil motion, respectively. At pre-shock-buffet conditions the flow response is predominantly at the airfoil motion frequency, with some smaller response at multiplies of this frequency. At shock buffet conditions, a key effect of prescribed airfoil motions on the buffeting flow is to create the possibility of a lock-in phenomenon, in which the shock buffet frequency is synchronized to the prescribed airfoil motion frequency for certain combinations of airfoil motion frequencies and amplitudes. Aerodynamic gain-phase models for the lock-in region, as well as for the pre-shock-buffet conditions are suggested, and also a possible relationship between the lock-in mechanism and limit-cycle oscillation is discussed.     

Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding

High-frequency limit-cycle oscillations of an airfoil at low Reynolds number are studied numerically. This regime is characterized by large apparent-mass effects and intermittent shedding of leading-edge vortices. Under these conditions, leading-edge vortex shedding has been shown to result in favorable consequences such as high lift and efficiencies in propulsion/power extraction, thus motivating this study. The aerodynamic model used in the aeroelastic framework is a potential-flow-based discrete-vortex method, augmented with intermittent leading-edge vortex shedding based on a leading-edge suction parameter reaching a critical value. This model has been validated extensively in the regime under consideration and is computationally cheap in comparison with Navier–Stokes solvers. The structural model used has degrees of freedom in pitch and plunge, and allows for large amplitudes and cubic stiffening. The aeroelastic framework developed in this paper is employed to undertake parametric studies which evaluate the impact of different types of nonlinearity. Structural configurations with pitch-to-plunge frequency ratios close to unity are considered, where the flutter speeds are lowest (ideal for power generation) and reduced frequencies are highest. The range of reduced frequencies studied is two to three times higher than most airfoil studies, a virtually unexplored regime. Aerodynamic nonlinearity resulting from intermittent leading-edge vortex shedding always causes a supercritical Hopf bifurcation, where limit-cycle oscillations occur at freestream velocities greater than the linear flutter speed. The variations in amplitude and frequency of limit-cycle oscillations as functions of aerodynamic and structural parameters are presented through the parametric studies. The excellent accuracy/cost balance offered by the methodology presented in this paper suggests that it could be successfully employed to investigate optimum setups for power harvesting in the low-Reynolds-number regime.     

Effect of Stress-softening on the Dynamics of a Load Supported by a Rubber String

The Mullins effect in the oscillatory motion of a load under gravity and attached to a stress-softening, neo-Hookean rubber string is investigated. Equations for the small amplitude vertical oscillations of the load superimposed on the finite static stretch of both the virgin and stress-softened cords, the latter subjected to varying degrees of preconditioning, are derived. The vibrational frequency of the small motion exhibits behavior similar to that observed in experiments by others on postmortem, human aortic tissue for which no stress-softening is reported. Standard numerical methods are applied to study the finite amplitude motion of the load in the stress-softened case. The resultant motions and their various physical aspects under free-fall and general initial conditions are described in several examples. Oscillations that engage all three phases of motion consisting of the suspension, the free-flight, and the retraction of the load in its general vertical motion are illustrated. Effects due to the degree of stress-softening are discussed; and the motion response for two values of the model softening parameter is compared in several examples. All results are illustrated graphically and numerous tabulated numerical results are provided.      

Viscous and inviscid interactions of an oblique shock with a flexible panel

The complex self-sustained oscillations arising from the interaction of an oblique shock with a flexible panel in both the inviscid and viscous regimes have been investigated numerically. The aeroelastic interactions are simulated using either the Euler or the full compressible Navier–Stokes equations coupled to the nonlinear von Karman plate equations. Results demonstrate that for a sufficiently strong shock limit-cycle oscillations emerge from either subcritical or supercritical bifurcations even in the absence of viscous separated flow effects. The critical dynamic pressure diminishes with increasing shock strength and can be much lower than that corresponding to standard panel flutter. Significant changes in panel dynamics were also found as a function of the shock impingement point and cavity pressure. For viscous laminar flow above the panel without a shock, high-frequency periodic oscillations appear due to the coupling of boundary-layer instabilities with high-mode flexural deflections. For a separated shock laminar boundary layer interaction, non-periodic self-excited oscillations arise which can result in a significant reduction in the extent of the time-averaged separation region. This finding suggests the potential use of an aeroelastically tailored flexible panel as a means of passive flow control. Forced panel oscillations, induced by a specified variable cavity pressure underneath the panel, were also found to be effective in reducing separation. For both inviscid and viscous interactions, the significant unsteadiness generated by the fluttering panel propagates along the complex reflected expansion/recompression wave system.     

鲁棒匹配点颤振预测的高效μ方法

基于含有参数不确定性的时域鲁棒颤振μ预测工具, 首先提出一种不确定多项式建模方法, 该方法通过线性分式变换(LFT)最终可以得到较低阶次不确定描述; 然后, 考虑到参数不确定问题中实μ(realμ)的计算复杂性, 又提出一种包含二分法的鲁棒颤振预测技术, 该方法是基于在一定飞行范围内飞行速度与气弹系统稳定性间的简单关系, 它能够避免鲁棒匹配点颤振预测中包含的高阶速度摄动块, 从而大大提高颤振预测的计算效率. 最后数值验证和对比表明了该方法的高效性.      

Nonlinear sound and structure interaction in a fluid–filled flexible waveguide

In this study, a 2-D infinite flexible waveguide is considered. The waveguide carries a weakly nonlinear acoustic fluid. It is bounded on one side by a weakly nonlinear flexible membrane and the other side is rigid. The infinite waveguide is driven at the origin by a piston oscillating at a single frequency. However, we focus only on the positive side of the piston. As the coupled waves propagate in the membrane and the fluid, the modal interactions lead to resonances and beats which form the main focus of this work. A regular perturbation method is used to derive the linear and the quasilinear order equations which are then solved. At the linear order, the primary wavenumbers are solved for and the modes are found to be non-orthogonal because of the flexible membrane. Only the propagating waves are included in the analysis. Both self-mode and cross-mode interactions of the planar and the non-planar modes are considered. The novelty of this work lies in obtaining conditions and the closed form solutions for the resonances and beats along the spatial coordinate. It is found that the self-mode interactions lead to beats only. And in the self-mode interactions, the coupled planar mode plays an important role. On the other hand, the cross-mode interactions can lead to either resonances or beats.     

Dry Friction Model of Percussive Drilling

It is postulated that the main mechanism of the enhancement of material removal rate (MRR) in percussive drilling is associated with generating impact forces, which act on the workpiece and help to develop micro-cracking in the cutting zone. The inherent non-linearity of the discontinuous impact process is modelled as a frictional pair, to generate the pattern of the impact forces. A novel formula for calculating the MRR is proposed, which explains the experimentally observed fall in MRR at higher static forces.     

Nonlinear Oscillations of Viscoelastic Rectangular Plates

Nonlinear oscillations of viscoelastic simply supported rectangular plates are studied by assuming the Voigt–Kelvin constitutive model. Using Hamilton's principle in conjunction with the kinematics associated with Kirchhoff's plate model, the governing equations of motion including the effect of damping are represented in terms of the transversal deflection and a stress function. Utilizing the Bubnov–Galerkin method, the nonlinear partial differential equations are reduced to an ordinary differential equation which is studied geometrically by approximate construction of the Poincaré maps. Explicit expressions are given for periodic solutions.     

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for strange phenomena to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szempliskia-Stupnicka, are then presented and compared to computer simulation results.     

Cross-well chaos and escape phenomena in driven oscillators

The paper is devoted to the study of common features in regular and strange behavior of the three classic dissipative softening type driven oscillators: (a) twin-well potential system, (b) single-well potential unsymmetric system and (c) single-well potential symmetric system.Computer simulations are followed by analytical approximations. It is shown that the mathematical techniques and physical concepts related to the theory of nonlinear oscillations are very useful in predicting bifurcations from regular, periodic responses to cross-well chaotic motions or to escape phenomena. The approximate analysis of periodic, resonant solutions and of period doubling or symmetry breaking instabilities in the Hill's type variational equation provides us with closed-form algebraic simple formulae; that is, the relationship between critical system parameter values, for which strange phenomena can be expected.     

非线性振动的一个新的渐近解法

本文在渐近法的基础上，引进谐波平衡思想，得到了一个新的渐近解法。与传统方法相比，应用本文方法求渐近解，不必解微分方程和依靠消除永年项建立补充工程，而是将求解过程转化为一系列的代数运算。因此，本文解法便于手算，更有利于用计算机计算高阶近似。     

Computational aeroelastic investigation of a transonic limit-cycle-oscillation experiment at a transport aircraft wing model

Aeroelastic measurements of a three-dimensional wing model, the so-called Aerostabil wing, were conducted in the Transonic Windtunnel Göttingen. This clean, backward-swept wing allowed the experimental investigation of limit cycle oscillations in a certain transonic parameter range. In this paper, a detailed insight into the observed physical phenomena, especially the measured limit cycle oscillations, is presented by means of CFD–CSM coupled simulations. These simulations on the basis of a detailed structural finite element model reveal the specific properties of the Aerostabil wing and furthermore allow investigating the unstable behavior of this windtunnel model for transonic flow settings. The aerodynamic characteristics include a two-shock system and large flow separation areas, further increasing the complexity of the aeroelastic problem. A structural single degree-of-freedom system is used for the prediction of the experimental stability range and the limit cycle oscillation investigations. Due to the good agreement of simulation and experiment the limit cycle oscillations can be explained by means of nonlinear aerodynamic effects.     

Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam

Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever beam–impactor systems. For off-resonance excitation at two and a half times the fundamental frequency, the response of the oscillating cantilever experiences period doubling as the separation distance or clearance between the beam axis and the contact surface is decreased. The nonlinear phenomenon is studied by using phase portraits, Poincaré sections, and spectral analysis. Motivated by atomic force microscopy, this general dynamic behavior is studied as a means to locating the separation distance corresponding to grazing where the contact force is minimized.     

On the dynamics of rotating circular cable structures

The combined effect of gravity and a centrifugal field on the dynamics of a heavy cable with negligible bending stiffness is treated. The inextensible length of the cable is preassigned in such a way that the cross-sectional shape of the cable structure is circular if the contiguration rotates about its central axis with a constant speed and no external forces act. Under the additional influence of gravity, complicated nonlinear weight-excited vibrations occur.To understand the variety of vibrational phenomena. The dynamic system is studied by some experiments first. In order to explain the experimental results theoretically, the governing nonlinear boundary value problem is derived next. Subsequently, an appropriate variational formulation for application of a Rayleigh Ritz procedure is suggested. The resulting nonlinear ordinary differential equations approximately deseribe a part of the observed vibrational behaviour.Both the experiments and the calculations demonstrate that, for high speeds, nonlinear weight-excited vibrations about the circular reference configuration occur. On the other hand, for low velocitites, periodie and even chaos-like snap-through phenomena appear.