An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system

We perform an analytical and experimental investigation into the dynamics of an aeroelastic system consisting of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The experimental results show that the onset of flutter takes place at a speed smaller than the one predicted by a quasi-steady aerodynamic approximation. On the other hand, the unsteady representation of the aerodynamic loads accurately predicts the experimental value. The linear analysis details the difference in both formulation and provides an explanation for this difference. Nonlinear analysis is then performed to identify the nonlinear coefficients of the pitch spring. The normal form of the Hopf bifurcation is then derived to characterize the type of instability. It is demonstrated that the instability of the considered aeroelastic system is supercritical as observed in the experiments.     

Slow oscillations of potential in electron-ion currents close to the emitter

This paper is concerned with quasi-stationary potential distributions in one-dimensional ion-electron currents close to an emitting surface. It is assumed that the part played by near collisions can be neglected. Three types of regimes unstable with respect to slow changes in the boundary conditions have been found. The instability is attributable to the feedback created by slow particles reflected to the emitter by potential barriers in the Debye layer close to the emitter.     

Self-sustained oscillations and bifurcations of transonic flow past simple airfoils

A turbulent flow past two symmetric airfoils, whose bow and aft portions are circular arcs, whereas midparts are flat, is studies numerically. The amplitude of lift coefficient oscillations versus the free-stream Mach number M _∞ is analyzed at zero angle of attack. Ranges of M _∞ in which there exist flow bifurcations are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 37–44, November–December, 2008     

On the large amplitude oscillations of a thin elastic beam

This paper is concerned with the finite amplitude, free, planar oscillations of a thin elastic beam. By assuming the motion to be inextensional but at the same time recognizing the existence of a resultant normal force acting on each cross-section of the beam a system of governing equations is derived which is manageable but still meaningful. For the case of the simply-supported beam a finite difference, Galerkin, and (regular) perturbation solutions are explicitly obtained. The results are compared and discussed. In the course of obtaining the various solutions it is found that an additional simplification in the form of the governing equations is possible. This simplification turns out to be quite important from a general point of view of obtaining approximate analytical solutions.     

The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes

Experiments were mounted to investigate the onset in a ‘Starling resistor’ of collapsible-tube oscillation, at the lowest possible Reynolds number so as to facilitate matched numerical simulations. The protocol adopted was to set pressure outside the tube and inside the tube at the upstream end, constant and equal to each other, then to progressively lower the downstream pressure past the point of tube collapse and, when this occurred, of oscillation onset. The working fluid was a glycerine/water mixture, and the silicone-rubber tube was suspended horizontally in air. Measurements were made of pressures and flow-rates and of the cross-sectional area at the approximate location of maximum oscillation; separately, the cross-sectional area of the tube in relation to transmural pressure was measured. Parameters varied in the flow experiments were the length of rigid pipe downstream of the collapsing tube, and the fluid viscosity. The pressure/flow-rate coordinates of both the point of peak flow-rate achieved before flow-rate limitation, and the point of oscillation onset, were satisfactorily independent of the pipe length downstream. Both points occurred at flow-rates that decreased with increasing fluid viscosity, so that the corresponding Reynolds numbers decreased more so. Oscillation did not break out below a Reynolds number of about 290 unless there was external mechanical agitation of the apparatus. The amplitude of oscillation decreased progressively towards zero at this point as viscosity was raised. After the flow-rate peak, flow limitation causes a local flow-rate minimum. When oscillation occurred, it started just before this minimum, and died away at the minimum.     

Unsteady transonic aerodynamic loadings on the airfoil caused by heaving,pitching oscillations and control surface

An implicit upwind finite volume solver for the Euler equations using the improved flux-splitting method is established and used to calculate the transonic flow past the airfoils with heaving, pitching oscillations and the control surface. Results are given for the NACA64A-10 airfoil which is in harmonic heaving and pitching oscillation and with the control surface in the transonic flow field. Some computational results are compared with the experiment data and the good agreements are shown in the paper.     

A describing function approach to the design of robust limit-cycle controllers

The design of robust limit-cycle controllers is introduced for autonomous systems with separable SISO nonlinearities. The objective is to design a controller to secure specified robust oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a Describing Function (DF) approach and then shaping the loop to reach desired limit-cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot. An example is given to illustrate the robustness of the controlled system to uncertainties in the linear subsystem model.     

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories.In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.     

Bifurcations and stability of amplitude modulated planar oscillations of an orbiting string with internal resonances

Nonlinear free transversal oscillations of an orbiting string satellite system are analyzed. They are governed by two partial integro-differential equations with quadratic nonlinearities. The system is weakly nonlinear but in practice works in conditions of nearly simultaneous internal resonance. The ability of truncated models to capture specific phenomena is discussed. By limiting the investigation to the planar motion with a one prevailing component perturbed out-of-plane, two different models with three modes in primary and secondary resonance are adopted. For increasing levels of the system energy, fundamental and bifurcated paths of fixed points are obtained and their stability is investigated. Moreover, periodically amplitude modulated planar motions and their stability for out-of-plane disturbances are studied.     

Sub/superharmonic oscillations and the perturbation procedure

The problem of sub/super and harmonic oscillations of non-linear systems is considered in this paper. This is made possible by the development of a modified version of the constrained Linstedt-Poincare perturbation procedure. The generality of the procedure is such that the existence of sub/super and harmonic oscillations can be simultaneously handled. Furthermore, the methodology developed is such that no a priori knowledge of solution form is necessary to establish the existence of such oscillation modes. Based on the approach, the steady response of Duffing's equation wherein sub/super and harmonic oscillations exist is studied in detail.     

Optimization of the damping decrement of free oscillations of a viscoelastic layered sphere with a limitation on weight

The problem of synthesis of a multilayer spherical shell with maximum damping of natural oscillations from a finite set of viscoelastic materials is considered with a limitation on weight. The necessary conditions of optimality are obtained, a computational algorithm is derived, and an example of calculation is presented. Institute of the Physiotechnical Problems of the North, Siberian Division, Russian Academy of Sciences, Yakutsk 677891. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 161–165, March–April, 2000.     

Cryogenic transonic wind tunnels and the condensation of nitrogen

A brief tutorial on the need for Reynolds number similarity and the advent of cryogenic transonic wind tunnels is presented. Experimental results of nitrogen condensation in nozzles are collected and related to the flow in the wind tunnels. New theoretical approaches to a solution of the condensation problem in the supersaturated state are proposed.List of symbols a speed of sound - A area - {ovc} wing area/wing span - c _p pressure coefficient, Eq. (12) - G ^* energy of formation of a critical droplet, Eq. (14) - h altitude - J homogeneous nucleation rate, Eq. (13) - k Boltzmann constant - l characteristic length - M Mach number, Eq. (2) - n ^* number of molecules in a critical droplet, Eq. (14) - p static pressure - p ₀ wind tunnel supply pressure - p ⁰ standard pressure - p equilibrium vapor pressure - P wind tunnel fan power - q dynamic pressure - Re Reynolds number, Eq. (1) - t time - T temperature - T ₀ wind tunnel supply temperature - molecular volume - V air speed - ratio of specific heats - dynamic viscosity - v kinematic viscosity - density - surface tension This paper is dedicated to my old friend Eberhard Berger upon his retirement from the Föttinger Institut of the Technical University of Berlin     

Hyperchaotic beats and their collapse to the quasiperiodic oscillations

The letter shows the possibility of generation of hyperchaotic beats characterized by four, three or two positive Lyapunov exponents. The beats are a result of linear coupling of two identical nonlinear subsystems describing second-harmonic generation of light (SHG). The rapid transition from highly chaotic beats to quasiperiodic oscillations is studied.     

Multiple solutions and stability of the steady transonic small-disturbance equation

Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.     

Numerical investigation of the effects of structural geometric and material nonlinearities on limit-cycle oscillation of a cropped delta wing

This article presents numerical simulations of the limit-cycle oscillation (LCO) of a cropped delta wing in order to investigate the effects of structural geometric and material nonlinearities on aeroelastic behavior. In the computational model, the structural part included both the geometric nonlinearity that arises from large deflections, and the material nonlinearity that originates from plasticity. The Euler equations were employed in the fluid part to describe the transonic aerodynamics. Moreover, the load transfer was conducted using a 3-D interpolating procedure, and the interfaces between the structural and aerodynamic domains were constructed in the form of an exact match. The flutter and LCO behaviors of the cropped delta wing were simulated using the coupling model, and the results were compared with existing experimental measurements. For lower dynamic pressures, the geometric nonlinearity provided the proper mechanism for the development of the LCO, and the numerical results correlated with the experimental values. For higher dynamic pressures, the material nonlinearity led to a rapid rise in the LCO amplitude, and the simulated varying trend was consistent with the experimental observation. This study demonstrated that the LCO of the cropped delta wing was not only closely related to geometric nonlinearity, but was also remarkably affected by material nonlinearity.     

Aerodynamic drag of bodies in transonic flow. Theory and applications to computational aerodynamics

Formulas for all the components of the aerodynamic drag (total, friction, inductive, wave, pressure, and heat-transfer) are uniformly derived as applied to flows governed by the Navier-Stokes and Reynolds equations. For flows of this type the definition of the aerodynamic drag components is refined and the physical basis of the chosen method of breaking up the total drag into components is discussed. Ways of calculating the aerodynamic drag components using the methods of computational aerodynamics are considered. On the basis of the refined formulas the drag components are calculated for flows around airfoils and a high-aspect-ratio wing in transonic flow.