A THEORETICAL MODEL FOR NONLINEAR PLANAR MOTIONS OF ROTORS UNDER FLUID CONFINEMENT

Following previous papers by Axisa, Antunes and co-workers, the authors address a theoretical model for immersed rotors, under moderate confinement, using simplified flow equations on the gap-averaged fluctuating quantities. However, in contrast to our previous efforts, the nonlinear terms of the flow equations are here fully accounted. Because such nonlinear analysis is quite involved, this paper will focus on the simpler case of planar motions, in order to emphasize the main aspects of our approach. A direct integration of the continuity and momentum equations leads to extremely lengthy formulations. Here, in order to solve the flow equations, we perform an exact integration of the continuity equation and an approximate solution of the momentum equation, based on a Fourier representation of the azimuthal pressure gradient. Then, an exact formulation for the dynamic flow force can be obtained. Our solution is discussed in connection with physical phenomena. Numerical simulations of the nonlinear rotor-flow coupled system are presented, showing that the linearized and the fully nonlinear models produces similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from these models are quite distinct. Experimental results indicate that the nonlinear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability and for linearly unstable regimes.     

An improved linear model for rotors subject to dissipative annular flows

In a previous paper, Antunes, Axisa, and co-workers, developed a linearized model for the dynamic of rotors under moderate fluid confinement, based on classical perturbation analysis, covering two different cases: (i) dissipative motions of a centered rotor; (ii) motions of an eccentric rotor for frictionless flow. Following the same procedures and assumptions, we derive here an improved model for the more general case of dissipative linearized motions of an eccentric rotor. Besides the rotor motion variables, a new variable—which can be interpreted as the fluctuating term of the average tangential velocity—is introduced, yielding an additional eigenvalue in the linear analysis. The new variable introduced, which is coupled with the rotor motions, is very convenient when frictional effects are not neglected. Under dissipative flows, a richer modal behavior is highlighted, which can be related to delay effects of the flow responses to the rotor motions. Our approach can be applied as well to other flow-excited systems, for example, those subjected to axial or leakage flows. Because rotor-dynamics are strongly dependent on the mean rotor eccentricity, the adequacy of this (or any other) model rely on using the actual value for such parameter.     

Nonlinear dynamics of flexible tethered satellite system subject to space environment

The paper studies the nonlinear dynamics of a flexible tethered satellite system subject to space environments, such as the J ₂ perturbation, the air drag force, the solar pressure, the heating effect, and the orbital eccentricity. The flexible tether is modeled as a series of lumped masses and viscoelastic dampers so that a finite multidegree-of-freedom nonlinear system is obtained. The stability of equilibrium positions of the nonlinear system is then analyzed via a simplified two-degree-freedom model in an orbital reference frame. In-plane motions of the tethered satellite system are studied numerically, taking the space environments into account. A large number of numerical simulations show that the flexible tethered satellite system displays nonlinear dynamic characteristics, such as bifurcations, quasi-periodic oscillations, and chaotic motions.     

An investigation on lateral and torsional coupled vibrations of high power density PMSM rotor caused by electromagnetic excitation

Electromagnetic excitation in high power density permanent magnet synchronous motors (PMSMs) due to eccentricity is a significant concern in industry; however, the treatment of lateral and torsional coupled vibrations caused by electromagnetic excitation is rarely addressed, yet it is very important for evaluating the stability of dynamic rotor vibrations. This study focuses on an analytical method for analyzing the stability of coupled lateral/torsional vibrations in rotor systems caused by electromagnetic excitation in a PMSM. An electromechanically coupled lateral/torsional dynamic model of a PMSM Jeffcott rotor is derived using a Lagrange–Maxwell approach. Equilibrium stability was analyzed using a linearized matrix of the equation describing the system. The stability criteria of coupled torsional–lateral motions are provided, and the influences of the electromagnetic and mechanical parameters on mechanical vibration stability and nonlinear behavior were investigated. These results provide better understanding of the nonlinear response of an eccentric PMSM rotor system and are beneficial for controlling and diagnosing eccentricity.     

Vibration characteristics of a hydraulic generator unit rotor system with parallel misalignment and rub-impact

The object of this research aims at the hydraulic generator unit rotor system. According to fault problems of the generator rotor local rubbing caused by the parallel misalignment and mass eccentricity, a dynamic model for the rotor system coupled with misalignment and rub-impact is established. The dynamic behaviors of this system are investigated using numerical integral method, as the parallel misalignment, mass eccentricity and bearing stiffness vary. The nonlinear dynamic responses of the generator rotor and turbine rotor with coupling faults are analyzed by means of bifurcation diagrams, Poincaré maps, axis orbits, time histories and amplitude spectrum diagrams. Various nonlinear phenomena in the system, such as periodic, three-periodic and quasi-periodic motions, are studied with the change of the parallel misalignment. The results reveal that vibration characteristics of the rotor system with coupling faults are extremely complex and there are some low frequencies with large amplitude in the 0.3–0.4× components. As the increase in mass eccentricity, the interval of nonperiodic motions will be continuously moved forward. It suggests that the reduction in mass eccentricity or increase in bearing stiffness could preclude nonlinear vibration. These might provide some important theory references for safety operating and exact identification of the faults in rotating machinery.     

Chaotic motions of a rigid rotor in short journal bearings

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature C radial clearance of bearing (m) - D diameter of bearing (m) - e dimensional eccentricity of journal (m) - e _s value of e corresponding to the static position of the journal - E dimensional static unbalance of rotor (m) - f _x, f _y =F _x/(P), F _y/(P): non-dimensional components of fluid film force - F _x, F _y dimensional components of fluid film force (N) - g acceleration of gravity (m/s²) - L axial length of bearing (m) - m % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor - M half mass of rotor (kg) - n angular speed of rotor (in r.p.m.=60/2) - t time     

Nonlinear analysis of a rub-impact rotor-bearing system with initial permanent rotor bow

A general model of a rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given. The nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory. The rubbing model is assumed to consist of the radial elastic impact and the tangential Coulomb type of friction. Through numerical calculation, rotating speeds, initial permanent bow lengths and phase angles between the mass eccentricity direction and the rotor permanent bow direction are used as control parameters to investigate their effect on the rub-impact rotor-bearing system with the help of bifurcation diagrams, Lyapunov exponents, Poincaré maps, frequency spectrums and orbit maps. Complicated motions, such as periodic, quasi-periodic even chaotic vibrations, are observed. Under the influence of the initial permanent bow, different routes to chaos are found and the speed when the rub happens is changed greatly. Corresponding results can be used to diagnose the rub-impact fault in this kind of rotor systems and this study may contribute to a further understanding of the nonlinear dynamics of such a rub-impact rotor-bearing system with initial permanent bow.     

碰摩裂纹转子轴承系统的周期运动稳定性及实验研究

根据碰摩裂纹耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶法，研究了系统周期运动的稳定性。研究发现，小偏心量下系统周期运动发生Hopf分岔，大偏心量下系统周期运动发生倍周期分岔，偏心量的加大使周期解的稳定性明显降低；系统碰摩间隙变小，碰摩影响了油膜涡动的形成，使失稳转速有所提高；裂纹深度的加大降低了系统周期运动的稳定性。本文的研究为转子轴承系统的安全稳定运行提供了理论参考。     

Dynamic response of a rub-impact rotor system under axial thrust

A model of a rigid rotor system under axial thrust with rotor-to-stator is developed based on the classic impact theory and is analyzed by the Lagrangian dynamics. The rubbing condition is modeled using the elastic impact-contact idealization, which consists of normal and tangential forces at the rotor-to-stator contact point. Mass eccentricity and rotating speed are used as control parameters to simulate the response of rotor system. The motions of periodic, quasi-periodic and chaotic are found in the rotor system response. Mass eccentricity plays an important role in creating chaotic phenomena.     

Vibration control of helicopter blade flapping via time-delay absorber

In this study, the controller is used to suppress the vibration due to rotor the helicopter blade flapping motion. The objective of this paper is to investigate the effect of time-delay absorber on the vibrating system when subjected to multi-parametric excitation forces. The equations of motion are described by coupled nonlinear differential equations. The averaging method is applied to obtain the frequency response equations near simultaneous sub-harmonic and internal resonance. The stability of the obtained nonlinear solution is studied and solved numerically. Numerical simulations show the steady state response amplitude versus the detuning parameter and the effects of the parameters system and controller. Effectiveness of the absorber E _a is about 2.7×10⁵ of the main system (X).     

多自由度转子-轴承系统周期运动分岔及稳定性研究

建立考虑诸多因素的转子-轴承系统多自由度模型,将与Newmark结合的打靶法应用到多自由度转子-轴承系统的周期稳定性分析中。着重研究了转子-轴承系统失稳转速随系统偏心量、轴承间隙、润滑油动力粘度以及轴承长径比的变化规律,研究结果表明:提高系统偏心量、减小轴承间隙、增大润滑油动力粘度以及选择适当的轴承长径比均能提高转子-轴承系统的失稳转速;对于不同的参数值,系统表现出不同的分岔规律,系统发生半速涡动时表现为倍周期分岔或拟周期分岔,发生油膜振荡时则表现为拟周期分岔。     

Pressure and velocity of air during drying and storage of cereal grains

A common method of drying cereal grains is to ventilate a large static mass of grain with an even flow of air at near ambient temperature. After the grain has been dried it is often stored in the same container and kept cool by aeration with a lower velocity of air than is used in drying. To analyse the airflow through this mass of grain a nonlinear momentum equation for flow through porous media is used where the resistance to flow is a + b ¦ν¦. This equation, together with the assumption that the air is incompressible, defines the problem which is solved numerically, using the finite element method, and the results compared with experimental values. The small parameter ε = bν _r /a, where ν _r is the velocity scale, is used in a perturbation analysis to examine the nonlinear effects of the resistance on the airflow. When ε = 0 the equations reduce to those for potential flow, while for small values of ε there are first-order corrections to the pressure p ₁ and the stream function χ ₁. The nonlinear problem is simplified by changing to curvilinear coordinates (s, t) where s is constant on the potential flow isobars while t is constant on the streamlines. General conclusions are derived for p ₁ and χ ₁, for example that they depend on the curvature of the potential flow solution with a large curvature of the isobars leading to larger values of p ₁ and similarly for the streamlines. The potential flow solution p ₀ and the first order solution p ₀ + εp ₁ are close to the solution of the full nonlinear problem when ε is small. To illustrate this for a typical grain storage problem, the solution p ₀ is shown to be very close to the finite element solution (with a difference of less than 1%) when ε < 0.03 while for the first order solution p ₀ + εp ₁ the difference is less than 1% when ε < 0.1.     

Turbocharger rotor dynamics with foundation excitation

To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed.     

Rheological–dynamical analogy: Prediction of damping parameters of hysteresis damper

This research aims to predict the damping parameters of hysteresis damper based on an analytical rheological–dynamical (RDA) visco-elasto-plastic solution of one-dimensional longitudinal continuous vibrations of a bar. A visco-elasto-plastic bar or damper is an energy dissipation device. An attempt is made to estimate quantitatively the influence of material physical parameters of materials on the damping ratio in both the linear visco-elastic analysis and the nonlinear visco-elasto-plastic analysis of damper subjected to external vibration forces. Two types of damping are considered: viscous damping in the case of linear analysis, defined as stiffness and/or mass proportional and, in the case of nonlinear analysis, hysteresis damping caused by inelastic deformations of damper. Owing to the visco-elastic nature of the materials of the damper and the frequency dependence of the viscous damping ratio ξ, it is useful to consider separately the situations arising when ξ is positive (the system is stable) and when it is negative. A negative damping ratio means that the complementary solution of the response would not die away (the system is unstable because of factor e^{ξ · ω · t}). In the case of nonlinear analysis, the force–displacement relation is nonlinear, so it is very difficult to predict the actual damping and stiffness coefficients, even if the force–displacement characteristic is simply perfect elasto-plastic. Using the RDA method, which takes into account the rate of release of visco-elasto-plastic energy of the dissipation devices; nonlinear behaviors are linearized, enabling to obtain the equivalent damping and stiffness coefficients and the effective period for the damper.     

Model representations of the interaction between the solar wind and the supersonic interstellar medium flow. prediction and interpretation of experimental data

The planning and conducting of physical experiments requires the development of theoretical models capable either of predicting possible experimental data or explaining those already obtained. The processes taking place in the physical world can be understood only in terms of the close interaction between theory and experiment. Developing any quantitative or qualitative model of a physical phenomenon requires a mathematical apparatus, on the basis of which such models can be constructed. The branch of theoretical science using the methods of magnetohydrodynamics and hydroaeromechanics for studying space physics problems is usually called cosmic gasdynamics; it is mostly used in developing models of physical phenomena occurring under space conditions. In order to emphasize the importance of cosmic gasdynamics in the development of astrophysics and space research, we will present several examples of models constructed by aerodynamicists. These models not only played an important role in qualitative predictions but are still being developed due to the need for the quantitative interpretation of the experimental data. The solar corona was long thought to be a formation in a state of gravitational equilibrium (Chapman model). However, it turned out that the pressure at infinity obtained on the basis of this equilibrium solution is considerably greater than the estimated pressure in the interstellar gas surrounding the solar corona. In [1] it was concluded that in this case the solar corona gas must expand and a solution describing this expansion was obtained by invoking the steady-state hydrodynamics equations in the spherically-symmetric approximation. The solution of these equations led to the theoretical prediction of the solar wind, a radial flow of fully ionized hydrogen plasma issuing from the solar corona at a low subsonic velocity but already hypersonic at the Earth’s orbit. Subsonic-to-supersonic transition is ensured by solar gravitation which in this case plays the role of a convergent-divergent nozzle. Within a year, the theoretical prediction of the solar wind [1] was confirmed by its experimental detection [2] onboard the Soviet spacecraft Luna-2. It turned out that at the Earth’s orbit the mean velocity of the solar wind V _E ≈ 450 km·s^?1, the mean proton temperature T _E ≈ 6 · 10⁴ K (the electron temperature is somewhat higher), and the mean concentration of protons (and electrons) n _E ≈ 10 cm^?3. The first hydrodynamic model of the supersonic solar-wind flow past the Earth’s magnetosphere [3] was only qualitative, since it considered a flow past a plane magnetic dipole in the approximation of a thin layer between the bow shock and an “obstacle” embedded in the flow. However, it was constructed before the actual discovery of the solar wind and provided further important impetus to the development of models of the supersonic solar wind flow past planets with a detached shock. One more example is furnished by the gasdynamicmodel of the solar wind flow past cometary atmospheres, first suggested in In this work, a model of the interaction between the supersonic solar wind and the supersonic flow of the local, i.e., surrounding the Sun, interstellar medium is considered; it was first suggested in [6] in a much simplified formulation. This model has been actively developed in connection with the flights of the spacecraft Voyager 1 and 2, Ulysses, Hubble Space Telescope, SOHO, and others, exploring the outer regions of the solar system.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

On unimodular constitutive expressions

We consider constitutive expressions which the stress σ(X, t) at a particle X at time t is given by σ (X, t) = $\text{F}$[F[X, τ)] where $\text{F}$[F(X, τ)] denotes a functional of the history of the deformation gradient matrix [F(X, τ)] from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant under the group of unimodular transformations, i.e. $\text{F}$[F(X, τ)] = $\text{F}$[F(X, τ) H] must hold for all matrices H such that det H - 1. We employ results from the classical theory of invariants in order to determine the general form of the expression $\text{F}$[F(X, τ)] which is consistent with these restrictions. Special cases are considered where the functional is replaced by a function of the strain, rate of strain, ? matrices. The case of shear flow is briefly discussed.     

Proving conditional attractivity of a closed set with a family of liapunov functions

Considering a closed set M of some x-space and a solution x(t), y(t) of a differential system x = X(x, y, t), y = Y(x, y, t), we give sufficient conditions in order that x(t) approaches M. We use several auxiliary functions and employ Salvadori's method of a one parameter family of Liapunov functions. An application is given to the two-body problem in the presence of some friction forces and when the reference frame is non-inertial.     

Asymptotic Stability of N-Solitary Waves of the FPU Lattices

We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Bäcklund transformation and use the result to analyze the linearized FPU equation.