Bifurcation analysis for a modified Jeffcott rotor with bearing clearances

A HB (Harmonic Balance)/AFT (Alternating Frequency/Time) technique is developed to obtain synchronous and subsynchronous whirling motions of a horizontal Jeffcott rotor with bearing clearances. The method utilizes an explicit Jacobian form for the iterative process which guarantees convergence at all parameter values. The method is shown to constitute a robust and accurate numerical scheme for the analysis of two dimensional nonlinear rotor problems. The stability analysis of the steady-state motions is obtained using perturbed equations about the periodic motions. The Floquet multipliers of the associated Monodromy matrix are determined using a new discrete HB/AFT method. Flip bifurcation boundaries were obtained which facilitated detection of possible rotor chaotic (irregular) motion as parameters of the system are changed. Quasi-periodic motion is also shown to occur as a result of a secondary Hopf bifurcation due to increase of the destabilizing cross-coupling stiffness coefficients in the rotor model.     

Characterization of the tip clearance flow in an axial compressor using 3-D digital PIV

The accurate characterization and simulation of rotor tip clearance flows has received much attention in recent years due to their impact on compressor performance and stability. At NASA Glenn the first known three dimensional digital particle image velocimetry (DPIV) measurements of the tip region of a low speed compressor rotor have been acquired to characterize the behavior of the rotor tip clearance flow. The measurements were acquired phase-locked to the rotor position so that changes in the tip clearance vortex position relative to the rotor blade can be seen. The DPIV technique allows the magnitude and relative contributions of both the asynchronous motions of a coherent structure and the temporal unsteadiness to be evaluated. Comparison of measurements taken at the peak efficiency and at near stall operating conditions characterizes the mean position of the clearance vortex and the changes in the unsteady behavior of the vortex with blade loading. Comparisons of the 3-D DPIV measurements at the compressor design point to a 3D steady N-S solution are also done to assess the fidelity of steady, single-passage simulations to model an unsteady flow field.     

On the beneficial effect of rotor suspension anisotropy on viscous-dry hysteretic instability

The destabilizing influence of the internal friction on the supercritical rotor whirl can be efficiently counterbalanced by other external dissipative sources and/or anisotropic suspension systems. The theoretical approach may take the internal dissipation into consideration either by dry or viscous models. Nevertheless, several numerical results and a new perturbation technique of the averaging type prove that similar rotor motions and stability limits are achievable by both models, whence the linear viscous assumption appears preferable. Thus, the internal hysteretic force may be expressed by the product of an equivalent viscous coefficient and the rotor centre velocity relative to a reference frame rotating with the shaft end sections. After calculating the natural frequencies and the response to dynamic imbalances, the stability of the steady motion is checked by the Routh-Hurwitz criterion, focusing the analysis on the individual influence of several characteristic properties, like the gyro structure, the stiffness anisotropy of the supports and their asymmetry, and searching for the external damping level needed for stability. A?fairly interesting result is that the benefit of the suspension anisotropy is most effective for a symmetric rotor mounted at the shaft mid-span and decreases significantly on increasing the configuration asymmetry, even moderately. It is also observed how the stability may somehow be associated with the coupling between progressive and retrograde precession motions.     

Stability of finite-amplitude and overstable convection of a conducting fluid through fixed porous bed

The stability of infinitestimal steady and oscillatory motions and finite amplitude steady motions of a conducting fluid through porous media with free boundaries which is heated from below and cooled from above is investigated in the presence of a uniform magnetic field. Infinitesimal steady motions are investigated using Liapunov method and its is shown that the principle of exchange of stability is valid only when P_m/P_r≤1 with a restricted value of the Hartmann number. It is shown that overstable motions are due to the zonal current induced by the magnetic field. Finite amplitude steady motions are investigated using Veronis [1] analysis and it is shown that for a restricted range of Hartmann numbers and porous parameter P_l, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than that value corresponding to oscillatory motions. Since the Busse number is greater than the wave number the horizontal scale of the steady finite-amplitude motions is larger than that of the overstable motions.     

Aeroelastic analysis of helicopter rotor blade in hover using an efficient reduced-order aerodynamic model

This paper presents a coupled flap–lag–torsion aeroelastic stability analysis and response of a hingeless helicopter blade in the hovering flight condition. The boundary element method based on the wake eigenvalues is used for the prediction of unsteady airloads of the rotor blade. The aeroelastic equations of motion of the rotor blade are derived by Galerkin's method. To obtain the aeroelastic stability and response, the governing nonlinear equations of motion are linearized about the nonlinear steady equilibrium positions using small perturbation theory. The equilibrium deflections are calculated through the iterative Newton–Raphson method. Numerical results comprising steady equilibrium state deflections, aeroelastic eigenvalues and time history response about these states for a two-bladed rotor are presented, and some of them are compared with those obtained from a two-dimensional quasi-steady strip aerodynamic theory. Also, the effect of the number of aerodynamic eigenmodes is investigated. The results show that the three-dimensional aerodynamic formulation has considerable impact on the determination of both the equilibrium condition and lead-lag instability.     

Stabilization of the Rotation Axis of a Solid by Coupled Perfectly Rigid Bodies

The problem of stabilizing the axis of a solid by coupled perfectly rigid bodies (PRBs) is solved. The solid executes a plane-parallel motion. The PRBs can rotate as a single rigid body about the centroidal axis of the solid and counterrotate about its transverse axes through equal angles. There is a particle inside the solid which causes its imbalance. It is established that the principal state (if any) of the system—rotation about the centroidal axis—is stable, whereas the rest (unwanted) states are unstable __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 122–129, August 2005.     

Steady motions in controlled gyrostat systems

Summary The equations of motion of a gyrostat system are given for the general case in which gyroscopic terms are present which are not due to the rotor motion. The stability of steady motions is investigated for free and for controlled systems. Observability and controllability are discussed, with the relative rotor speed as observation and the torque on the rotor as control variable.

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Stationäre Bewegungen geregelter Gyrostaten-Systeme

Übersicht Die Bewegungsgleichungen von Gyrostaten-Systemen werden für den allgemeinen Fall angegeben, in dem gyroskopische Terme auftreten, die nicht von der Rotorbewegung stammen. Die Stabilität stationärer Bewegungen des freien und des geregelten Gyrostaten werden untersucht. Steuerbarkeit und Beobachtbarkeit werden ebenfalls behandelt mit der relativen Rotordrehzahl als Beobachtung und dem Drehmoment um die Rotorachse als Steuergröße.

     

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.     

Steady motion of a rigid disk of finite thickness on a horizontal plane

The article discusses the steady motion of a rigid disk of finite thickness rolling on its edge on a horizontal plane under the influence of gravity. The governing equations are presented and two cases allowing for a steady-state solution are considered: rolling on consistently rough ground and rolling on perfectly smooth ground. The conditions of steady motion are derived for both kinds of ground and it is shown that the possible steady motion of a disk is either on a straight line or in a circle. Oscillations about steady state are discussed and conditions for stable motion established. The bifurcations of steady motions on a smooth surface are also considered.     

Influence of periodic excitation on self-sustained vibrations of one disk rotors in arbitrary length journals bearings

Interaction of forced and self-sustained vibrations of one disk rotor is described by nonlinear finite-degree-of-freedom dynamical system. The shaft of the rotor is supported by two journal bearings. The combination of the shooting technique and the continuation algorithm is used to study the rotor periodic vibrations. The Floquet multipliers are calculated to analyze periodic vibrations stability. The results of periodic motions analysis are shown on the frequency response. The quasi-periodic motions are investigated. These nonlinear vibrations coexist with the periodic forced vibrations.     

On steady motions of a rigid body bearing a two-degree-of-freedom control momentum gyroscope with precession axis arbitrarily oriented in the carrying body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has 8, 12, or 16 steady motions and the number of stable motions is 2 or 4.     

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.     

Stability analysis of steady rotations of a rigid body bearing two-degree-of-freedom control moment gyros with dissipation in gimbal suspension axes

To study the stability of steady rotations of a control moment gyro system with internal dissipation, we use the Barbashin-Krasovskii theorem and the relation, established in [1], between the Lyapunov function and steady motions. Taking into account the special properties of the original problem, we reduce it to a lower-dimensional problem.We give a detailed presentation of an algorithm for analyzing the stability of steady motions of a gyrostat and use this algorithm to perform a complete study for two systems consisting, respectively, of one and two gyros whose gimbal axes are parallel to the principal axis of inertia of the system. Each steady motion is identified as either asymptotically stable or unstable. We find periodic motions that exist only in the presence of dynamic symmetry and which are regular precessions. For the system with two gyros, we prove the asymptotic stability of quiescent states and prove that in the angular momentum range where these states are defined the system does not have any other stable motions.     

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

In previous papers, Antunes and co-workers developed a theoretical model for nonlinear planar motions—motions X (t) taking place in one single direction—of rotors under fluid confinement using simplified flow equations on the gap-averaged fluctuating quantities. The nonlinear solution obtained was shown to be consistent with a linearized solution for the same problem. Also, it displayed an encouraging qualitative agreement between the nonlinear theory and preliminary experimental results. Following a similar approach, the nonlinear theoretical model is here extended to cope with orbital rotor motions—motions X (t) and Y (t) taking place in two different orthogonal directions—by developing an exact formulation for the two- dimensional dynamic flow forces. Numerical simulations of the nonlinear rotor–flow coupled system are presented and compared with the linearized model. These yield similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from the linearized and nonlinear models may be quite distinct. Preliminary experimental results also 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.     

Grazing Bifurcation in the Response of Cracked Jeffcott Rotor

A cracked rotor is modeled by a piecewise linear system due to thebreath of crack in a rotating shaft. The differential equations ofmotion for the nonsmooth system are derived and solved with thenumerical integration method. From the simulation results, it isobserved that a grazing bifurcation exists in the response. Thegrazing bifurcation can give rise to jumps between periodic motions,quasi-periodic motions from the periodic ones, chaos, and intermittentchaos.     

On limit motions of a system of control moment gyros with intrinsic dissipation in a homogeneous gravitational field

We study the limit motions of a free rigid body bearing n two-degree-of-freedom control moment gyros with dissipation in the gyro gimbal suspension axes. We show that, in the absence of dynamic symmetry, the limit motions of the system are only steady rotations at a constant angular velocity. In the case of dynamic symmetry, the gyros can be arranged so that, in addition to steady rotations, the system exhibits limit motions that are regular precessions.     

Non-periodic motions of a Jeffcott rotor with non-linear elastic restoring forces

The conditions that give rise to non-periodic motions of a Jeffcott rotor in the presence of non-linear elastic restoring forces are examined. It is well known that non-periodic behaviours that characterise the dynamics of a rotor are fundamentally a consequence of two aspects: the non-linearity of the hydrodynamic forces in the lubricated bearings of the supports and the non-linearity that affects the elastic restoring forces in the shaft of the rotor. In the present research the analysis was restricted to the influence of the non-linearity that characterises the elastic restoring forces in the shaft, adopting a system that was selected the simplest as possible. This system was represented by a Jeffcott rotor with a shaft of mass that was negligible respect to the one of the disk, and supported with ball bearings. In order to check in a straightforward manner the non-linearity of the system and to confirm the results obtained through theoretical analysis, an investigation was carried out using an experimental model consisting of a rotating disk fitted in the middle of a piano wire pulled taut at its ends but leaving the tension adjustable. The adopted length/diameter ratio was high enough to assume the wire itself was perfectly flexible while its mass was negligible compared to that of the disk. Under such hypotheses the motion of the disk centre can be expressed by means of two ordinary, non-linear and coupled differential equations. The conditions that make the above motion non-periodic or chaotic were found through numerical integration of the equations of motion. A number of numerical trials were carried out using a 4th order Runge-Kutta routine with adaptive stepsize control. This procedure made it possible to plot the trajectories of the disk centre and the phase diagrams of the component motions, taken along two orthogonal coordinate axes, with their projections of the Poincaré sections. On the basis of the theoretical results obtained, the conditions that give rise to non-periodic motions of the experimental rotor were identified.     

转子系统振动变参控制中的瞬态响应

本文以可变参数的挤压油膜阻尼器作为控制元件，研究了转子在稳态转速及加速运动过程中进行变参控制时的瞬态响应问题。结果说明了对转子系统的振动进行分段变参控制，无论是在稳态还是在加速运动过程中，一般都可以取得满意的控制效果，不仅可以减小转子系统的振动，而且还可以使转子系统平稳地通过具有较大振动的共振区，但变参位置不应在多值转速区内。     

碰摩转子系统的非光滑分析

通过建立转子系统碰摩的Ｐｏｉｎｃａｒｅ映射,将对非光滑碰摩系统的研究转化为对Ｐｏｉｎｃａｒｅ映射的分析,文中主要对转子碰摩当中一类特殊的运动形式－单点碰摩下的擦边现象者了详细研究。从序列的极限理论出发分析了该映射的周期不动点的稳定性及其吸引域,得到了转子系统在接近擦边运动时解随系统参数变化的分岔情形。