Nonlinear Oscillations of a Nonresonant Cable under In-Plane Excitation with a Longitudinal Control

The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.     

Control of the aerodynamic characteristics of a profile oscillating with respect to the incidence angle in subsonic viscous flow

The results of a numerical simulation of the unsteady subsonic viscous gas flow around a two-dimensional profile oscillating with respect to the incidence angle are presented and the possibility of controlling the nonstationary aerodynamic characteristics is considered. The hysteresis phenomena typical of oscillatory profile motions are investigated, the dependence of the lift force and drag is found for various laws of periodic variation of the incidence angle with time, and the effect of the frequency and amplitude of the angular profile oscillations on the shape of the hysteresis curves is studied. The calculations were based on the numerical solution of the nonstationary Navier-Stokes equations averaged in the Reynolds sense (Reynolds equations) which were closed using the k-ω turbulence model with modeling of the laminar/turbulent transition.     

Oscillations of a cylinder piercing a linearly stratified fluid layer

The plane problem of the small steady-state oscillations of a horizontal cylinder arbitrarily located in a three-layer fluid whose upper and lower layers are homogeneous and whose middle layer is linearly stratified is considered in the linear formulation using the Boussinesq approximation. The fluid is assumed to be ideal and incompressible. The method of mass sources distributed along the body contour is used in the internal wave generation regime and an integral equation for the fluid pressure is derived in the non-wave regime. The hydrodynamic load acting on the body is calculated as a function of the oscillation frequency of the cylinder and its location. The results are compared with experimental data.     

Skin friction and heat transfer in rapidly oscillating boundary layers

The influence of rapid oscillations in the outer part of a boundary layer upon the time-averaged skin friction and heat transfer is investigated analytically. The oscillations are taken to be harmonic. The only restriction on the oscillation amplitude is that it should be sufficiently small to permit the use of the boundary layer equations. The derived asymptotic formulae show the explicit dependence of momentum transfer on the frequency and the time-averaged boundary layer flow. For the heat transfer similar formulae can be derived in a number of limiting cases, viz. when the Prandtl number is either large or small.     

Non-axisymmetric oscillations of a hemispherical drop

Natural oscillations of a hemispherical drop on a solid substrate are considered. The Hocking condition is used to take into account the contact angle dynamics. The natural oscillations attenuate due to dissipation on the contact line and the degeneracy in the azimuthal number is eliminated. Forced oscillations of the drop are studied for tangential vibrations of the substrate. For the case of a fixed contact line, the amplitude of the forced oscillations grows without bound near the fundamental requency. In other cases, the amplitude is finite.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 8–20. Original Russian Text Copyright © 2004 by Lyubimov, Lyubimova, Shklyaev.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Natural oscillations of a rotating spherical fluid layer of variable depth

Small harmonic oscillations of the free surface of a thin fluid layer covering a rotating sphere are considered. The fluid is in the central field of sphere gravity and is exposed to the centrifugal and Coriolis forces. It is assumed that the fluid layer depth is independent of the longitude. In this formulation the problem is governed by a differential equation with singular coefficients that generalizes the Laplace tidal equation. The method of local separation of singularities is applied to integrate this equation. The solutions obtained are compared with the corresponding modes of the Laplace tidal equation, that is, the solutions of the problem for a fluid layer of constant depth.     

Turbulent Channel Flow with an Artificial Two-Dimensional Wall Layer

Turbulent flow of an incompressible fluid in a plane channel with parallel walls is considered. The three-dimensional time-dependent Navier-Stokes equations are solved numerically using the spectral finite-difference method. An artificial force which completely suppresses lateral oscillations of the velocity is introduced in the near-wall zone (10 % of the channel half-width in the neighborhood of each wall). Thus, the three-dimensional flow zone, in which turbulent oscillations can develop, is separated from the wall by a fluid layer. It is found that the elimination of three-dimensionality in the neighborhood of the walls leads to a significant reduction in the drag. However, complete laminarization does not occur. The flow in the stream core remains turbulent and can be interpreted as a turbulent flow in a channel with walls located on the boundary of the two-dimensional layer and traveling at the local mean-flow velocity. The oscillations developing inside the two-dimensional layer, which have significant amplitude, distort the flow only in the adjacent zone. Beyond this zone the distributions of the mean characteristics and the structure of instantaneous fields completely correspond to ordinary turbulent flow in a channel with rigid walls. The results obtained confirm the hypothesis of the unimportance of the no-slip boundary conditions for the fluctuating velocity component in the mechanism of onset and self-maintenance of turbulence in wall flows.     

An improved nonlinear reduced-order modeling for transonic aeroelastic systems

In this study, an improved nonlinear reduced-order model composed of a linear part and a nonlinear part is explored for transonic aeroelastic systems. The linear part is identified via the eigensystem realization algorithm and the nonlinear part is obtained via the Levenberg–Marquardt algorithm. The impulsive signal is chosen as the training signal for the linear part and the sinusoidal signal is used to determine the order of the linear part. The training signal for the nonlinear part is selected as the filtered white Gaussian noise with the maximal amplitude and frequency range to be designed via the aeroelastic responses. An NACA64A010 airfoil and an NACA0012 airfoil are taken as illustrative examples to demonstrate the performance of the presented reduced-order model in modeling transonic aerodynamic forces. The aeroelastic behaviors of the two airfoils are obtained via computational fluid dynamics to solve the Euler equation and the Navier–Stokes equation, respectively. The numerical results demonstrate that the presented reduced-order model can successfully predict the nonlinear aerodynamic forces with and without viscous flows. Moreover, the presented reduced-order model is capable of capturing the flutter velocity and modeling complex aeroelastic behaviors, including limit-cycle oscillations, beat phenomena and nodal-shaped oscillations at the transonic Mach numbers with high accuracy.     

Oscillation behavior of solutions for even order neutral functional differential equations

Even order neutral functional differential equations are considered. Sufficient conditions for the oscillation behavior of solutions for this differential equation are presented. The new results are presented and some examples are also given.     

Various methods of reconstruction of a cavity in an orthotropic layer

Direct and inverse problems of oscillations of an anisotropic layer with a cylindrical cavity of an arbitrary cross-sectional shape under the action of a load applied to the layer surface are considered. An asymptotic approach to solving these problems with cavities of small relative sizes is proposed. Numerical results of solving direct and inverse problems are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 181–189, May–June, 2009.     

Experimental Study of Self-Oscillations Developing in a Swirling-Jet Flow

The results of an experimental investigation of the periodic pressure oscillations developing in a submerged swirling jet flowing out into an open space are presented. The main governing parameters of the process are determined and a general dependence relating the oscillation frequency with the governing parameters over a wide range of parameter variation is established experimentally. On the basis of this dependence, an empirical formula for calculating the oscillation frequency is proposed.     

Oscillations of conservative systems with complex trajectories

The principles of formation of closed and quasiperiodic orbitally stable trajectories of conservative systems are formulated. It is revealed that irregular oscillations are due to the orbital instability of quasiperiodic oscillations. A bistable oscillator with periodic forcing is considered. The existence of random oscillations at low level of energy and the conditions for orbitally stable oscillations are established To the Beginning of the Third Millennium __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 3–25, July 2008.     

Stability in a model of a delayed neural network

The stability of the null solution in a system of coupled cells is investigated. Each cell evolves according to Hopfield's equation for an analog circuit, with a delay incorporated to account for finite switching speed of amplifiers. A necessary and sufficient condition on the connection matrix is obtained for delay-induced oscillations to be possible in a general (not necessarily symmetric) network.     

Radiative and convective heat transfer in hypersonic flow around a blunt body

The heat transfer in the vicinity of the critical point is investigated for hypersonic air flow around a blunt body. The gas-dynamical conservation equations are solved simultaneously with the radiative transport equation in integral form. Allowance is made for the viscosity, heat conduction, and the actual radiation parameters of air, including spectral line emission. Profiles are obtained for the thermodynamic variables along the critical line. The dependence of the radiative and convective components of the aerodynamic heating on the velocity and pressure ahead of the shock front as well as the radius of curvature of the blunt nose section is discussed. Approximate relations having the form of similarity laws are derived for the heat fluxes in the vicinity of the critical point. The limits of applicability of the thermodynamic equilibrium approximation in the shock-compressed layer are discussed. The influence of absorption of radiation from the compressed layer by the cold freestream on the aerodynamic heating is considered. Attention is given in this case to the dependence of the spectral absorption coefficient for the cold air on the intensity of the radiation incident upon it.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–123, September–October, 1972.     

Calculation of the longitudinal velocity in the linear problem of a vibrator in a subsonic boundary layer

The linear problem is considered of a localized vibrator mounted on a flat plate in a subsonic boundary layer. The plate and the vibrator are assumed to be heat-insulated, and the dimensions of the vibrator and the frequency of the oscillations are such that the flow may be described by means of the equations of a boundary layer with self-induced pressure. The amplitude of the oscillations of the vibrator and the perturbations of the flow parameters corresponding to it are assumed to be small, and this makes it possible to linearize these equations. Integral transformations are used to construct a solution for values of the time greatly exceeding the period of the oscillations of the vibrator. The profiles of the perturbations of the longitudinal velocity are calculated in dependence on the transverse coordinate for various values of the longitudinal coordinate. A comparison is made with the profiles of the perturbations of the longitudinal velocity which have been obtained experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1987.     

Influence of Adsorption and Desorption of Gas Micronuclei on the Nature of the Flow of a Gassy Liquid through a Porous Medium

The problem of gassy liquid flow through a porous medium is considered theoretically. Periodic oscillations of the liquid and gas flow rate observed experimentally are attributable to the processes of sorption and desorption of gas micronuclei on the walls of the pore space and their diffusion. In the kinetic equation employed the desorption rate is directly proportional to the adsorbed micronucleus concentration and the seepage rate, and the adsorption rate is directly proportional to the product of the mobile micronucleus concentration and the free site concentration on the pore surfaces. Steady-state solutions of this equation are investigated. It is shown that periodic oscillations of the flow rate can manifest themselves only when the processes of micronuclei adsorption predominate over the desorption processes.     

Interaction of torsion and lateral bending in aircraft nose landing gear shimmy

In this paper we consider the onset of shimmy oscillations of an aircraft nose landing gear. To this end we develop and study a mathematical model with torsional and lateral bending modes that are coupled through a wheel-mounted elastic tyre. The geometric effects of a positive rake angle are fully incorporated into the resulting five-dimensional ordinary differential equation model. A bifurcation analysis in terms of the forward velocity and the vertical force on the gear reveals routes to different types of shimmy oscillations. In particular, we find regions of stable torsional and stable lateral shimmy oscillations, as well as transient quasiperiodic shimmy where both modes are excited.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.     

On chaotic motions of systems with dry friction

The principles of formation of closed and quasiperiodic orbitally stable trajectories in systems with dry friction are formulated. A frictional oscillator with periodic forcing is considered. It is established that random and stable oscillations exist. It is discovered that orbital instability and irregular oscillations are associated with the instability domain near a singular point Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 123–132, September 2008.