Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the non- linearity of the airfoil section＇s freeplay. There are two crit- ical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincar6 map is constructed for the limit cycle os- cillations by using piecewise-linear solutions with and with- out contact in the system. Through analysis of the Poincar6 map, a series of equations which can determine the frequen- cies of period-1 limit cycle oscillations at any flight veloc- ity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agree- ment is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. More- over, there exist multi-periods limit cycle oscillations and even complicated ＂chaotic＂ oscillations may occur, which are usually found in smooth nonlinear dynamic systems.

The Axial Nonlinear Vibration Analysis of Ball-screw about Machine Tool Feeding System

The forced state of the ball-screw of machine tool feeding system is analyzed. The ball-screw is simplified as Timoshenko beam and the differential equation of motion for the ball-screw is built. To obtain the axial vibration equation,the differential equation of motion is simplified using the assumed mode method. Axial vibration equation is in form of Duffing equation and has the characteristics of nonlinearity. The numerical simulation of Duffing equation is proceeded by MATLAB / Simulink. The effect of screw length,exciting force and damping coefficient are researched,and the axial vibration phase track diagram and Poincare section are obtained. The stability and period of the axial vibration are analyzed. The limit cycle of phase track diagram is enclosed. Axial vibration has two type-center singularity distributions on both sides of the origin. The singularity attracts vibration to reach a stable state,and Poincare section shows that axial vibration appears chaotic motion and quasi periodic motion or periodic motion. Singularity position changes with the vibration system parameters,while the distribution doesn’ t change. The period of the vibration is enhanced with increasing frequency and damping coefficient. Test of the feeding system ball-screw axial vibration exists chaos movement. This paper provides a certain theoretical basis for the dynamic characteristic analysis of machine feeding system ball-screw and optimization of structural parameters.

The dynamics of an ion acting on two monochromatic obliquely propagating Alfvén waves

The interaction between a magnetized ion and two monochromatic shear Alfvtn waves propagating obliquely to an ambient magnetic field is investigated both analytically and numerically. According to the Hamiltonian of this system, the invariant of motion at the lowest order and the half-island widths at the corresponding resonances are obtained analytically using the Lie transformation method. It is shown that these theoretical results agree with the numerical ones from the Poincare surface of section. The regular motions from the invariant and the transition to stochasticity due to resonance overlapping are demonstrated. Compared to the case of a single wave, there may be a lower stochastic threshold in the multiple-wave problem.

Analysis of two-torus in a new four-dimensional autonomous system

In this paper, we report the dynamical behaviours of a four-dimenslonal autonomous continuous dissipative system analysed when the parameter is varied in the range we are interested in. The system changes its dynamical modes between periodic motion and quasiperiodic motion. Furthermore, the existence of two-torus is investigated numerically by means of Lyapunov exponents. By taking advantage of phase portraits and Poincaré sections, two types of the two-torus are observed and proved to have the structure of ring torus and horn torus, both of which are known to be the standard tori.

Chaos feature of test particle in the gravitational field with a quadrupole

In this paper we investigate the dynamics of a test particle in the gravitational field with a quadrupole. By constructing Poincare sections for different values of the parameters and initial conditions, we find a chaotic evolution.From these Poincare sections, we further confirm that the chaotic evolution of the test particle originates from the quadrupole.

THE DOUBLE MODE MODEL OF THE CHAOTIC MOTION FOR A LARGE DEFLECTION PLATE

The primary aim of this paper is to study the chaotic motion of a large deflection plate. Considered here is a buckled plate, which is simply supported and subjected to a lateral harmonic excitation. At first, the partial differential equation governing the transverse vibration of the plate is derived. Then, by means of the Galerkin approach, the partial differential equation is simplified into a set of two ordinary differential equations. It is proved that the double mode model is identical with the single mode model. The Melnikov method is used to give the approximate excitation thresholds for the occurrence of the chaotic vibration. Finally numerical computation is carried out.

Correspondence between classical dynamics and recurrence spectra of Rydberg hydrogen atom near a metal surface

The chaotic behaviours of the Rydberg hydrogen atom near a metal surface are presented. A numerical comparison of Poincare surfaces of section with recurrence spectra for a few selected scaled energies indicates the correspondence between classical motion and quantum properties of an excited electron. Both results demonstrate that the scaled energy dominates sensitively the dynamical properties of system. There exists a critical scaled energy εc, for ε 〈 εc, the system is near-integrable, and as the decrease of ε the spectrum is gradually rendered regular and finally turns into a pure Coulomb field situation. On the contrary, if ε 〉 εc, with the increase of ε, the system tends to be non-integrable, the ergodic motion in phase space presages that chaotic motion appears, and more and more electrons are adsorbed on the metal surface, thus the spectrum becomes gradually simple.

The dynamical properties of a Rydberg hydrogen atom between two parallel metal surfaces

This paper presents the dynamical properties of a Rydberg hydrogen atom between two metal surfaces using phase space analysis methods. The dynamical behaviour of the excited hydrogen atom depends sensitively on the atom-surface distance d. There exists a critical atom-surface distance dc = 1586 a.u. When the atom-surface distance d is larger than the critical distance de, the image charge potential is less important than the Coulomb potential, the system is near-integrable and the electron motion is regular. As the distance d decreases, the system will tend to be non-integrable and unstable, and the electron might be captured by the metal surfaces.

Evolution of Periodic Orbits in the Sun-Saturn System

We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitational Sun-Saturn system incorporating actual oblateness of Saturn in the planar circular restricted three body problem. In this paper, we study the effect of solar radiation pressure on the location of Sun centered and Saturn centered orbits, its diameter, semi major axis and eccentricity by taking different values of solar radiation pressure q and different values of Jacobi constant “C”, and by considering actual oblateness of Saturn using Poincare surface of section (PSS) method. It is ob-served that by the introduction of perturbing force due to solar radiation pressure admissible range of Jacobi constant C decreases, it is also observed that as value of C decreases the number of islands decreases and as a result the number of periodic and quasi periodic orbits decreases.Fur-ther, the periodic orbits around Saturn and Sun moves towards Sun by decreasing perturbation due to solar radiation pressure q for a specific choice of Jacobi constant C. It is also observed that due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces, whereas, due to solar radiation pressure uniform change in semi major axis and eccen-tricity of Saturn centered periodic orbits is observed.

Analysis of Effect of Oblateness of Smaller Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of oblateness of smaller primary on these orbits are considered. It is observed that oblateness of smaller primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to planetary oblateness has to be understood and should be taken care of during trajectory design. In this paper, detailed stability analysis of periodic orbit having three loops is given for A2 = 0.0001.

Analysis of Effect of Solar Radiation Pressure of Bigger Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of solar radiation pressure of bigger primary and actual oblateness of smaller primary on these orbits areconsidered. It is observed that solar radiation pressure of bigger primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to solar radiation pressure has to be understood and should be taken care of during trajectory design. It is also verified that stability of such orbits are negligible so they can be used as transfer orbit. For each pair of solar radiation pressure q and Jacobi constant C we get two separatrices where stability of island becomes zero. In this paper, detailed stability analysis of periodic orbit having two loops is given when q = 0.9845.

Evolution of the “f” Family Orbits in the Photo Gravitational Sun-Saturn System with Oblateness

We analyze the periodic orbits of “f” family (simply symmetric retrograde periodic orbits) and the regions of quasi-periodic motion around Saturn in the photo gravitational Sun-Saturn system in the framework of planar circular restricted three-body problem with oblateness. The location, nature and size of these orbits are studied using the numerical technique of Poincare surface of sections (PSS). In this paper we analyze these orbits for different solar radiation pressure (q) and actual oblateness coefficient of Sun Saturn system. It is observed that as Jacobi constant (C) increases, the number of islands in the PSS and consequently the number of periodic and quasi-periodic orbits increase. The periodic orbits around Saturn move towards the Sun with decrease in solar radiation pressure for given value of “C”. It is observed that as the perturbation due to solar radiation pressure decreases, the two separatrices come closer to each other and also come closer to Saturn. It is found that the eccentricity and semi major axis of periodic orbits at both separatrices are increased by perturbation due to solar radiation pressure.

An improved numerical method for constructing Halo/Lissajous orbits in a full solar system model

An improved numerical method that can construct Halo/Lissajous orbits in the vicinity of collinear libration points in a full solar system model is investigated. A full solar system gravitational model in the geocentric rotating coordinate system with a clear presentation of the angular velocity relative to the inertial coordinate system is proposed. An alternative way to determine patch points in the multiple shooting method is provided based on a dynamical analysis with Poincare′sections. By employing the new patch points and sequential quadratic programming, Halo orbits for L1, L2, and L3points as well as Lissajous orbits for L1and L2points in the EarthMoon system are generated with the proposed full solar system gravitational model to verify the effectiveness of the proposed method.

Statistical properties of quantum spectra in nuclei

Some aspects of quantum chaos in a finite system have been studied based on the analysis of statistical behavior of quantum spectra in nuclei.The experiment data show the transition from order to chaos with increasing excitation energy in spherical nuclei.The dependence of the order to chaos transition on nuclear deformation and nuclear rotating is described.The influence of pairing effect on the order to chaos transition is also discussed.Some important experiment phenomena in nuclear physics have been understood from the point of view of the interplay between order and chaos.

Effect of intramolecular energy transfer on the rate of unimolecular isomerization: HCN --HNC

This paper reports calculations of the rate of isomerization of HCN - HCN based on the theory of Gary and Rice as extended by Zhao and Rice. The major task is to determine the effect of intramolecular energy transfer on the prediction of the rate of isomerization. Both the full three-dimensional (3D) system and the reduced two-dimensional (2D) system obtained from freezing CN bond at 1.159 A are analyzed to check the validity of the freezing bond approximation. Meanwhile, RRKM rates are calculated to test RRKM choice of the transition state by comparing to Gary-Rice three-state model. The comparison shows that the rates from 2D model and 3D model are differing up to 20% with 2D rates consistently larger. The intramolecular energy transfer modifies the isomerization rate for HCN system up to 30% that is modestly small by the expectation. The isomerization rate predicted from RRKM theory is greater than those of Gary-Rice three-state model theory up to 65%, and it overstimates the rates under all considerations, including the contribution from intramolecular energy transfer, for the studied system by about a factor of 2.