Nonlinear dynamics of a statically misaligned flexible rotor in active magnetic bearings

The response of a statically misaligned flexible rotor mounted in active magnetic bearings is numerically investigated in this work. The mathematical model of the rotor-bearing system incorporates nonlinearity due to the geometric coupling of the magnetic actuators as well as that arising from the magnetic actuator forces that are nonlinear function of the coil current and the air gap between the rotor and the stator. The influence of the rotor’s static misalignment, represented by the gravity parameter, W, on its response was found to be dependent on the magnitude of the geometric coupling parameter, α. Numerical results showed that for α = 0, the response of the rotor was always synchronous regardless of the values of W. For moderate values of α, nonsynchronous vibration was seen in the response of the rotor for the case of W ≠ 0. For large values of α, nonsynchronous vibration was observed in the response of the rotor irrespective of the values of W. For the values of design and operating parameters of the rotor-bearing system investigated in this work, the response of the rotor displayed a rich variety of nonlinear dynamical phenomena including sub-synchronous vibrations of period-2, -3, -4, -6, -8, -12, -14 and -16, quasi-periodicity and chaos. Numerical results further revealed the existence of multiple attractors within certain ranges of the speed parameter, Ω. Co-existence of attractors has serious implications on the safe operation of magnetically supported rotating machinery as synchronous response of the rotor may become nonsynchronous or even chaotic when excited by external forces that cause the rotor’s position to move from one basin of attraction to another.     

Chaos via torus breakdown in the vibration response of a rigid rotor supported by active magnetic bearings

This work reports on a numerical study undertaken to investigate the response of an imbalanced rigid rotor supported by active magnetic bearings. The mathematical model of the rotor-bearing system used in this study incorporates nonlinearity arising from the electromagnetic force—coil current—air gap relationship, and the effects of geometrical cross-coupling. The response of the rotor is observed to exhibit a rich variety of dynamical behavior including synchronous, sub-synchronous, quasi-periodic and chaotic vibrations. The transition from synchronous rotor response to chaos is via the torus breakdown route. As the rotor imbalance magnitude is increased, the synchronous rotor response undergoes a secondary Hopf bifurcation resulting in quasi-periodic vibration, which is characterized by a torus attractor. With further increase in the rotor imbalance magnitude, this attractor is seen to develop wrinkles and becomes unstable resulting in a fractal torus attractor. The fractal torus is eventually destroyed as the rotor imbalance magnitude is further increased. Quasi-periodic and frequency-locked sub-synchronous vibrations are seen to appear and disappear alternately before the emergence of chaos in the response of the rotor. The magnitude of rotor imbalance where sub-synchronous, quasi-periodic and chaotic vibrations are observed in this study, albeit being higher than the specified imbalance level for rotating machinery, may possibly occur due to a gradual degradation of the rotor balance quality during operation.     

Non-linear dynamic analysis of a flexible rotor supported on porous oil journal bearings

In the present paper, the non-linear dynamic analysis of a flexible rotor with a rigid disk under unbalance excitation mounted on porous oil journal bearings at the two ends is carried out. The system equation of motion is obtained by finite element formulation of Timoshenko beam and the disk. The non-linear oil-film forces are calculated from the solution of the modified Reynolds equation simultaneously with Darcy’s equation. The system equation of motion is then solved by the Wilson-θ method. Bifurcation diagrams, Poincaré maps, time response, journal trajectories, FFT-spectrum, etc. are obtained to study the non-linear dynamics of the rotor-bearing system. The effect of various non-dimensional rotor-bearing parameters on the bifurcation characteristics of the system is studied. It is shown that the system undergoes Hopf bifurcation as the speed increases. Further, slenderness ratio, material properties of the rotor, ratio of disk mass to shaft mass and permeability of the porous bush are shown to have profound effect on the bifurcation characteristics of the rotor-bearing system.     

Chaotic response and bifurcation analysis of a flexible rotor supported by porous and non-porous bearings with nonlinear suspension

This study presents numerical work investigating the dynamic responses of a flexible rotor supported by porous journal bearings. Both porous and non-porous bearing types are taken into consideration in this study. The rotating speed ratios and imbalance parameters are also presented and proved to be important control parameters. Many non-periodic responses to chaotic and quasi-periodic motions are found, too. From the bifurcation diagrams in this paper, it is also evidenced that the vibration behaviors would be improved by porous bearings. The modeling result obtained here can be employed to predict the dynamics of bearing–rotor systems, and undesirable behavior of the rotor and bearing orbits can be avoided. Also, this could help engineers and researchers in designing and studying bearing–rotor systems or some turbo-machinery in the future.     

A network approach to the modelling of active magnetic bearings

The modelling of active magnetic bearings based on a network approach is considered. Unlike in the standard modelling approach, where a linearization of the current-force relation for the centred shaft position is used, network models permit to include the position dependence of the bearing force in the force model. This becomes necessary when model based controllers are used to stabilize a magnetically supported shaft in tracking applications.

The approach is based on the well known application of network models to magnetic circuits. Further simplifying assumptions are discussed which allow one to obtain a network with a limited number of lumped parameters describing the magnetic behaviour of a magnetic bearing. The modelling of a combined radial and axial bearing serves as an example for the application of the proposed approach. Furthermore, the fitting of the network based model to measured characteristic force curves is discussed. In this context, a method for including saturation effects in the model is sketched.     

Forced oscillations of an unbalanced rotor in nonisotropic bearings

The influence of anisotropy of elastic bearings on forced oscillations of a rotor with the static and moment unbalance is studied for the cases of its fastening on a rigid shaft and on a flexible one. The rotor with four degrees of freedom is considered. It is suggested that the shaft is fixed in linear elastic nonisotropic bearings. The differential equations of rotation of the rotor are written in complex variables, and an exact solution to the equation system is found that corresponds to the elliptical synchronous precession. The exact solution is a sum of two vectors, one of which parameterizes a forward precession, while another parameterizes a reverse precession. Amplitude-frequency characteristics of forward and reverse precessions and elliptical trajectories of the rotor axis ends are constructed. It is shown that, in case of nonisotropic bearings, both the forward and reverse precession, as well as the axis motion of nonsimple type (when its one end is moving forward, while another is moving in the reverse direction), can take place. The influence of anisotropy of elastic bearings also manifests itself by change in critical frequencies towards their reduction and by arising of additional critical frequencies in the bottom part of the spectrum, which significantly complicates dynamics of the high-speed rotor at the moment when it reaches the working angular speed.     

Approximation of quasi-periodic solutions of a rotor in two-lobe bearings with time-varying geometry

Improving the dynamic behaviour of rotor systems in journal bearings represents an ongoing topic of research. The pressure distribution within journal bearings is described by the Reynolds equation, whereby unwanted oscillations can be caused by the fluid-solid interaction within the bearings. An approach of a two-lobe bearing with time-varying geometry is suggested to suppress or at least to reduce occurring oscillations. In order to systematically analyse the system, a spectral reduction is performed, allowing to handle also quasi-periodic behaviour by means of numerical continuation algorithms. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Whirl characteristics of a flexible liquid-filled rotor under thermal shock

In this paper, the whirl characteristics of a flexible liquid-filled rotor subjected to thermal shock are investigated. On the basis of the Hamilton principle, the whirl frequency equation of the rotor system is derived. Using Laplace transform, the analytical model of the temperature field of the rotor is obtained. The validity of the developed temperature model is demonstrated by comparing with the finite element results. Then, the thermal axial force exerted on the rotor is calculated and the influence factors are studied. The system stability is analyzed in terms of the whirl frequency equation. The reasonability of the predict model for system stability is verified, and a good agreement can be seen in the comparison of the obtained results based on the presented analytical method with published data. Finally, the critical spinning speed of the rotor system is analyzed, and the effects of some main parameters on system critical speed are investigated.     

Stability and vibration analysis of a complex flexible rotor bearing system

This paper presents the non-linear dynamic analysis of a flexible rotor having unbalanced and supported by ball bearings. The rolling element bearings are modeled as two degree of freedom elements where the kinematics of the rolling elements are taken into account, as well as the internal clearance and the Hertz contact non-linearity. In order to calculate the periodic response of this non-linear system, the harmonic balance method is used. This method is implemented with an exact condensation strategy to reduce the computational time. Moreover, the stability of the non-linear system is analyzed in the frequency-domain by a method based on a perturbation applied to the known harmonic solution in the time domain.     

Deformation of a flexible current-carrying plate in a constant magnetic field

A solution procedure is proposed for the deformation of current-carrying annular plates with nonlinear geometry in a constant magnetic field. We assume that the deformation of the plate by the time-dependent Lorentz force is a slow process and the inertial terms can be ignored. The problem is solved numerically by quasilinearization and discrete orthogonalization methods.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 67–70, 1988.     

Nonlinear response and dynamic stability of a cracked rotor

Establishment of a new approach for analyzing the nonlinear behavior of a cracked rotor system is the main goal of the present research. Nonlinear governing equations of motion are developed for the cracked rotor system with asymmetrical viscoelastic supports. In establishing the approach, the masses of the rotational shaft and a disc mounted on the shaft, geometric nonlinearity of the shaft, and the rotor’s extra displacements due to the existence of the crack are all taken into account. On the basis of the governing equations, the nonlinear behavior of the rotor system is analyzed numerically with considerations of the effects of the crack depth, the crack location, the locations of the disc, and the shaft’s rotational speed. The effects of the crack and the other system parameters on the dynamic stability of the rotor system are also investigated.     

Geometric effects on surface roughness in electrodeposition

^** Email: ranga{at}ufl.edu Our aim is to understand the early stages of roughness growthin electrodeposition. We present a study of the branching ofsteady planar electrode surfaces to steady non-planar electrodesurfaces as the voltage imposed at a cathode is decreased throughits critical value. We aim to discover the type of branchingand to learn whether or not the new branches can be detectedby measuring the current. The cross-sectional shape of the electrodemakes a difference. Circular electrodes lead to transcriticalbranching; ordinarily, rectangular electrodes lead to a pitchfork,sometimes forward and sometimes backward.     

Application of a hybrid method to the nonlinear dynamic analysis of a flexible rotor supported by a spherical gas-lubricated bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the nonlinear dynamic behavior of a flexible rotor supported by a spherical gas-lubricated bearing system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and quasi-periodic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those from other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of spherical gas film rotor–bearing systems.     

Bifurcation and nonlinear analysis of a flexible rotor supported by a relative short spherical gas bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the bifurcation and nonlinear dynamic behavior of a flexible rotor supported by a relative short spherical gas bearing (RSSGB) system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic, and chaotic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of RSSGB systems.     

The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features

The purpose of this paper is to investigate the coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features. There are some new phenomena due to the coupling effects of shaft-bending, shaft-torsion, disk-transverse and blade-bending. In this investigation, this paper mainly focuses on the influence of mistuned features of the blade's length and the stagger angle. It is found that there are four types of coupling modes: the coupling mode of shaft bending, disk transverse and blade bending (SDB), the coupling mode of shaft torsion disk transverse-blade bending (TDB), the coupling mode of disk transverse and blade bending (DB), the repeated mode of blade bending-blade bending (BB). With the effect of mistuned features, the natural frequencies and the coupling mode type will change correspondingly. With the mistuning value of blade length employed in this study, the TDB mode in the tuned system will disappear and shift into TSDB mode instead, and one of the repeated SDB modes will be replaced by STDB modes. Due to this mistuned features, the blades and disk experience a certain degree of vibration localization phenomenon. Different from the length feature, the influence of mistuning values of blade's stagger angle mainly take effect on the coupling modes. At last, by inspection on the Campbell diagrams, the influence of rotational speed on the transformation of natural frequencies is illustrated on the tuned/mistuned flexible shaft-disk-blades coupling structure.     

Dynamics of a self-sustained electromechanical system with flexible arm and cubic coupling

This paper studies the dynamics of a self-sustained electromechanical system with nonlinear coupling. The mechanical part is a flexible beam. The Krylov–Bogoliubov averaging method is used to derive oscillatory solutions. Focus is made on the effects of the detuning parameter and nonlinear coupling. The largest Lyapunov exponent is used to determine chaotic domains of the system.     

Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.     

Geometric quantization as zero mass limit in the problem of a particle on a sphere having a magnetic monopole at its center

The quantum-mechanical problem of a point particle on a sphere with a magnetic monopole at its center is shown to be equivalent in the zero mass limit to the quantum theory including geometric action related to the Kirillov—Konstant form for the SU(2) group.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 3–8, 1990.