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Vibration control in supercritical rotors is a challenging task due to the underlying complex dynamics that high-speed machinery undergoes. When taking into account both structural (e.g., structural integrity and material properties) and operational (e.g., rotational speed variation and environmental effects) uncertainties that can plague such systems, the task of designing and implementing a robust and reliable control system becomes increasingly daunting. In this contribution, a Kriging modeling approach is used to derive a representative surrogate-based controller applied to a flexible rotor supported by active magnetic bearings. The surrogate controller is formulated by considering the relationship between the shaft position and actuator current. This means that after proper training, the surrogate control takes the rotor position (proximity sensor readings) as input, and the output corresponds to the current levels for the actuators to control the system vibration amplitudes. Experimental tests considering a test rig composed of a flexible rotor supported by two active magnetic bearings were used to validate the proposed approach for different operational scenarios, namely, steady-state, run-up, and transient-state. In this case, the surrogate model was trained considering the shaft positions and the corresponding control currents measured with the rotor operating close to its first bending critical speed. The obtained results illustrate how the Kriging-based surrogate controller can reliably maintain acceptable vibration levels, even when exposed to unexpected constant rotation speeds and other operating conditions not manifested in previous sample sets. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear vibration of a flexible rotor supported on squeeze-film dampers (SFDs) with centering springs is analyzed using the incremental harmonic balance (IHB) method, and bifurcation phenomena appeared in the resonance region are investigated. Complex nonlinear phenomena occur in this system due to the interaction of the fluid-film forces and the unbalance forces of the rotor in the SFD. Systems with these complex nonlinearities cannot be solved using the classical IHB methods. To overcome this problem, the classical IHB method and the alternating frequency/time (AFT) method are combined. The processing of linear matrices is performed in the same way as the classical IHB method, and only the processing of nonlinear force matrix caused by fluid–structure interaction is modified (application of transformation matrix). To prove the validity of the proposed method, the results calculated using the proposed method are compared with the results calculated using the Runge–Kutta method and the results presented in reference. Then, frequency response curves according to changes in bearing parameter B , gravity parameter W , stiffness ratio K , mass ratio α , and unbalance parameter U are constructed. Stability and bifurcation analyses of the calculated solution are performed using the Floquet theory. The proposed method can be effectively applied to the nonlinear vibration analysis of rotor systems supported on fluid-film bearings. [ABSTRACT FROM AUTHOR]

Frequency-dependent adaptive noise cancellation-based tracking controller (ANC-TC) is a known technique for the stabilization of several nonlinear dynamical systems. In recent years, this control strategy has been introduced and applied for the stabilization of a flexible rotor supported on full-lubricated journal bearings. This paper proposes a theoretical investigation, based on robust immersion and invariance (I&I) theory, of a novel ANC-frequency estimation (FE) technique designed to stabilize a flexible rotor shaft affected by self-generated sinusoidal disturbances, generalized to the case of unknown frequency. A structural proof, under assumptions on closed-loop output signals, shows that the sinusoidal disturbance rejection is exponential. Numerical simulations are presented to validate the mathematical results in silico. The iterative Inexact Newton method is applied to the disturbance frequency and phase estimation error point series. The data fitting confirms that the phase estimation succession has an exponential convergence behavior and that the asymptotical frequency estimation is a warm-up phase of the overall close-loop disturbance estimation process. In two different operating conditions, the orders of convergence obtained by phase and frequency estimate timeseries are p φ = 1 ,   p ω , u n c = 0.9983 and p ω , c a v = 1.005 . Rejection of the rotor dynamic disturbance occurs approximately 76% before in the cavitated than in the uncavitated condition, 2 (s) and 8.5 (s), respectively. [ABSTRACT FROM AUTHOR]

Bearing skidding is a frequent phenomenon in rotating machinery, which causes equipment motion instability and bearing wear failure. Due to the difficulties in the manufacture of high-speed flexible rotor test rig, measurement of bearing motion parameters and complex structure of angular contact ball bearing (ACBB), which makes the research on double-piece inner ring ACBB skidding challenging. This paper studies the ACBB skidding mechanism from the aspect of rotor characteristics, and first finds the resonance skidding phenomenon. Firstly, a high-speed flexible rotor test rig is designed and built to study the ACBB skidding mechanism, and the influence of foreign matter on the bearing cage slip rate is systematically studied. Then, the bearing cage speed considering the influence of lubricating oil pollution is accurately measured based on the adaptive fractional short-time Fourier transform. Finally, bearing skidding diagnosis under variable working conditions is realized based on the deep meta-transfer learning with feature enhanced generative adversarial network and average deflection power threshold. These proposed strategies systematically solve some problems in the research field of ACBB skidding, which has high practical significance and theoretical guidance value. • The resonance skidding phenomenon is first found by considering flexible rotor characteristics. • Bearing cage speed is accurately measured based on the eddy current sensor. • The influence of foreign matter on ball bearing skidding is first systematically studied. • Bearing skidding diagnosis is realized based on the deep meta-transfer learning. [ABSTRACT FROM AUTHOR]

To achieve acceptable dynamical behavior for large rotating machines operating at subcritical speeds, the balancing quality check at the planned service speed in the installation location is often demanded for machines such as turbo-generators or high-speed machines. While most studies investigate the balancing quality at critical speeds, only a few studies have investigated this aspect using numerical methods at operational speed. This study proposes a novel, model-based method for inversely estimating initial residual unbalance in one and two planes after initial grade balancing for large flexible rotors operating at the service speeds. The method utilizes vibration measurements from two planes in any single direction, combined with a finite element model of the rotor to inversely determine the residual unbalance in one and two planes. This method can be practically used to determine the initial and residual unbalance after the balancing process, and further it can be used for condition-based monitoring of the unbalance state of the rotor. [ABSTRACT FROM AUTHOR]

Aiming at the problem that base excitation would cause the rotor to collide to protect the bearing and even cause the system to lose stability, the vibration displacement response characteristics and control of the active magnetic bearing-flexible rotor system under the condition of base translational excitation were studied. First, the motion equation of the active magnetic bearing-flexible rotor system in the base coordinate system was established. Then, an adaptive controller was designed with the base acceleration as the input signal and the LMS variable step algorithm, the output signal was adaptively adjusted according to the displacement difference, and the vibration displacement response of the system under this kind of disturbance was suppressed. A numerical simulation was carried out to analyze the vibration displacement response characteristics of the rotor system under the base translational excitation, and the control effect of the designed adaptive controller to control the vibration of the active magnetic bearing-flexible rotor system under the base translational excitation condition was studied. The results show that the vibration displacement response of the electromagnetic bearing-flexible rotor system under base translational excitation is related to the excitation frequency, excitation amplitude, excitation phase, etc. The adaptive controller proposed in this paper can effectively suppress the vibration of the active magnetic bearing-flexible rotor system under the base translational excitation. [ABSTRACT FROM AUTHOR]

An identification algorithm for the estimation of residual unbalances is developed for a flexible rotor-bearing system integrated with an active magnetic bearing (AMB). The main advantage of the present methodology is that it can be applied in situ and also with safe traversal of resonances during run-up. The flexible rotor-AMB system is modelled by the finite element method, considering randomly distributed residual unbalances along with disc discrete residual unbalances. In order to obtain equivalent discrete unbalances at limited disc locations for the balancing of the flexible rotor system, a generalized influence coefficient method (GICM) is proposed. Equivalent unbalances obtained at discrete balancing planes are used to check the reduction in unbalance responses. The proposed algorithm is compared with the conventional influence coefficient method (CICM) without AMB to check the accuracy and consistency of the estimated values of residual unbalances. The values estimated by both methods are found to be consistent, while the GICM has an advantage of using AMB for the safe operation of the unbalanced rotor at high speeds during measurements. In order to have a suitable discrete position of AMB (for effective estimation of residual unbalances), various rotor system models are considered with different positions of the AMB. [ABSTRACT FROM AUTHOR]

In modern industries, high-speed motors and generators have received great attention, and they are widely used in micro turbine, centrifugal compressor, blower, etc. However, the resonance vibration of flexible rotor will become a challenging issue when the rotor has to operate above the first bending critical speed. In this paper, a phase compensation method is proposed to improve the damping level of the flexible rotor around the first bending critical speed. The dynamic characteristics of the flexible rotor are analyzed, and the modal frequency is obtained. The rotor finite element model is verified by the modal test. Based on Proportion-Integration-Differentiation (PID) controller, the phase of the control system is shaped with different general filters to improve the damping level of the flexible rotor around the first bending critical speed. The simulation and experimental results indicate that the first bending mode damping of rotor is obviously enhanced by phase compensation. The phase compensation method can effectively suppress the resonance vibration of the rotor and make the rotor smoothly pass the first bending critical speed, achieving supercritical operation. [ABSTRACT FROM AUTHOR]

An algorithm for accounting for identification of the structural deviations impacting the dynamics of a flexible multispan rotor on full electromagnetic suspension in an actively developed computer model of dynamics of a flexible rotor on active magnetic bearings is presented. The algorithm is illustrated by applying it against the scale model of a rotor on active magnetic bearings. [ABSTRACT FROM AUTHOR]

In this paper, nonlinear dynamic analysis and experimental verification of a flexible rotor supported on the active magnetic bearings (AMB) are studied. The model of the system is formulated by eight degrees of freedom. This model takes in to account the gyroscopic moments of the disk, geometric coupling of the magnetic actuators and contact forces of the auxiliary bearings. The equations of motion are solved using the Rung–Kutta method. The effects of auxiliary bearings stiffness and rotational speed on the dynamic behavior of the system are investigated by the bifurcation diagrams, dynamic trajectories, power spectra analysis, Poincare´ maps and maximum Lyapunov exponent. In the experimental test rig, two special flexible supports are constructed that can adjust the required stiffness of the auxiliary bearings. The results indicate that the auxiliary bearings stiffness and rotational speed have significant effects on the dynamic responses and can be used as effective control parameters for the system. Good correlation between the experimental and theoretical results is found. Very rich forms of periodic, quasi-periodic, period −4 and chaotic vibrations are observed. The present study can be useful in designing and selection of suitable operating parameters. As a result, the system can avoid the undesirable behavior [ABSTRACT FROM AUTHOR]

Active magnetic bearings (AMBs) support rotors using electromagnetic forces rather than mechanical forces. It is necessary to identify the AMB force coefficients (equivalent stiffness and damping) since they play key roles in the rotordynamic analysis. The identification is usually performed by analyzing the unbalance response. However, the presence of unknown residual unbalances reduces the identification accuracy and a rigid rotor model is only valid when the rotating speed is far below the first bending critical speed. Therefore, this paper proposes an identification algorithm to estimate the stiffness and damping parameters for a flexible rotor AMB system in the presence of unknown residual unbalances by using two independent unbalance response data sets. The proposed algorithm is first evaluated by numerical calculation for accuracy and then applied experimentally in the identification of a flexible rotor AMB system operating at speeds ranging from 3,000 rpm (50 Hz) to 30,000 rpm (500 Hz). The identification results are verified in the end. [ABSTRACT FROM AUTHOR]

Rotor systems carried in transportation system or under seismic excitations are considered to have a moving base. To study the dynamic behavior of flexible rotor systems subjected to time-variable base motions, a general model is developed based on finite element method and Lagrange’s equation. Two groups of Euler angles are defined to describe the rotation of the rotor with respect to the base and that of the base with respect to the ground. It is found that the base rotations would cause nonlinearities in the model. To verify the proposed model, a novel test rig which could simulate the base angular-movement is designed. Dynamic experiments on a flexible rotor-bearing system with base angular motions are carried out. Based upon these, numerical simulations are conducted to further study the dynamic response of the flexible rotor under harmonic angular base motions. The effects of base angular amplitude, rotating speed and base frequency on response behaviors are discussed by means of FFT, waterfall, frequency response curve and orbits of the rotor. The FFT and waterfall plots of the disk horizontal and vertical vibrations are marked with multiplications of the base frequency and sum and difference tones of the rotating frequency and the base frequency. Their amplitudes will increase remarkably when they meet the whirling frequencies of the rotor system. [ABSTRACT FROM AUTHOR]

An algorithm for accounting for inner damping in an actively developed computer model of dynamics of a flexible rotor on active magnetic bearings is presented. The algorithm is illustrated by applying it for analyzing the effects of amplitude-independent inner damping on rotor dynamics when residual imbalance is present. [ABSTRACT FROM AUTHOR]

The resonance vibration control of active magnetic bearing (AMB) supported flexible rotors is a challenging topic in the industrial applications. This work addresses the two issues on the noncollocated AMB flexible rotor systems while passing through the critical speed. A modal separation scheme is established to enhance the observability of the bending mode, and a damping optimization procedure of the electromagnetic force is proposed to provide the maximum control efficiency. First, the rotor imbalance analysis is carried out, and several possible approaches to minimize the vibration of the flexible rotor are presented. Then, the detailed descriptions of the bending mode extraction scheme using notch filters and phase-lead compensators are discussed for a noncollocated flexible rotor system. The root locii of the closed-loop AMB-rotor system with and without additional phase-lead compensator are also performed. A solution to determining the phase angles between the measured rotor displacements to the bearing forces is established using the physical modeling and experimental identification. Finally, simulation and experimental results on a 10 kW magnetically suspended centrifugal compressor show the effectiveness of the proposed methods. [ABSTRACT FROM PUBLISHER]

A rotor supported by hydrodynamic bearings may undergo unstable motion and may exhibit several nonlinear phenomena in the vicinity of the critical stability speed. This paper presents a stability analysis of a flexible rotor supported by journal bearings using a nonlinear dynamic model and a short bearing approximation. Numerical continuation is applied to determine the boundaries of stability and the bifurcations of the limit cycles. Nonlinear phenomena such as jumping motion and bi-stability domain are predicted. An extended stability chart has also been determined including the domains of stable oscillatory motion. The investigation also includes the effect of rotor flexibility and bearing characteristics on the stability boundaries and on the safe operating speed range. For a selected range of bearing parameters, two Hopf bifurcation regions are found for high rotor stiffness, three regions for low stiffness and four bifurcation regions in transition between high and low stiffness. It has also been found that the stable operating speed range decreases with rotor flexibility and bearing parameter. [ABSTRACT FROM AUTHOR]

Classical traveling-wave ultrasonic motors (TWUMs) usually generate the elliptical clusters on the stator surface through the bending vibration of the annular stator on the basis of piezoelectric excitation, and the torque output and power transmission are realized by friction coupling under the rational preload. Stable and reliable working characteristics attracted the focus of various applications. High energy consumption and low efficiency are restricted factors for the service performance of motors. Displacement amplification and stress regulation based on flexible design become a possible approach to improve motor performances. In this study, a TWUM30 prototype was proposed with consideration of (sub) structure optimization, i.e., tooth height and backlash of the stator elastomer. The flexible design of the rotor structure was carried out for consideration of the contact area and stress regulation. After integrated bonding of the stator and PZT4, the electrical and vibration characteristics of the stator prototype are measured by an impedance analyzer and high-precision laser micro-displacement sensor. Results showed that the proposed TWUM30 is capable of operation under low-voltage driving (41.9 kHz @ 80 V p), and the extreme mechanical performances presented the stall toque of 64mN∙m and the no-load speed of 153.8 rpm. With the combination of the flexible design and systematic integrated manufacturing of the stator/rotor, low-voltage driving of the TWUM30 motor under ultra-low preload was realized at low cost and the adverse effects of excessive preload were suppressed. This study is expected to provide support for high reliability design and long-life service of the TWUMs. It also provides a realistic possibility for the flexible design and application of energy-saving and high-efficient traveling-wave piezoelectric drivers. [Display omitted] • TWUM30 prototype was proposed with (sub)structures optimization and integrated manufacturing. • Flexible redesign of rotor was conducted for the contact area and stress regulation. • TWUM30 is capable of operation under low preload and low-voltage driving (40 N, 80 V p @41.9 kHz). • Mechanical performances (toque & speed) of TWUM30 were compared and improved. [ABSTRACT FROM AUTHOR]

A new method of modal balancing for flexible rotors considering the impact of elastic supports is proposed since flexible rotors supported on elastic supports cannot be efficiently balanced using the current modal balancing method. A finite element model of a rotor is established and a rotor test rig is built up. After calculating and measuring critical speeds and vibration modes, modal balancing is conducted through the rotor testing system. Results suggest that the relative error between the calculated and measured critical speeds is only 0.36% and the modal assurance criterion of the first calculated and measured mode shapes is 0.990 6. The method of modal balancing for flexible rotors considering elastic supports is reasonable, the algorithms are correct and the number of trial mass can be reduced under the premise of ensuring fully orthogonal balance. A combination of calculated support stiffness, i.e., an elastic support at left end and a rigid support at right end, is accordant with the actual situation of rotor structure and the vibration reduction percentages from CH1 to CH3 reach more than 80% and CH4 up to 38%. As the overall damping effect is obvious, the modal balancing method considering elastic supports has a prospect in engineering application and it can provide technical reference for balance tests of flexible rotors in aero engines. [ABSTRACT FROM AUTHOR]

This paper proposes a phase compensator design for the flexible rotor in the high-energy-density magnetically suspended motor (HEDMSM), whose rated rotational speed is above the first bending critical speed. The optimum phase and magnitude of the control system nearby the critical speed are analyzed. Additionally, the coil inductance of the large-power HEDMSM is always large. That will lead to large phase lag and make optimum compensator design more difficult. Therefore, an effective phase-leading compensation method is presented to compensate the large phase lag of the switching power amplifier. Then, the optimum compensator is designed and carried out in the 315-kW HEDMSM. The experimental results show that the optimum compensator can minimize the displacement resonance peak, when the rotor passes the first bending critical speed with low amplifier current. [ABSTRACT FROM AUTHOR]

In order to minimizing the rotor displacement and the amplifier current mainly caused by the unbalance forces when the flexible rotor passes the first bending critical speed, the optimal controller is presented in this paper. The accurate modeling method for the flexible rotor based on the sine sweeping measurements is investigated. The design of the Kalman estimator and the choice of the variance matrix elements have been described. The optimal state feedback regulator with an integral controller has been used for stabilizing the system and the determination of the weight matrices has been investigated in detail. The influences of the specific elements of the weight matrices on the resonance peak of the flexible rotor when passing the first bending critical speed are analyzed. Finally, the running up test of the flexible rotor is implemented and the result shows the effectiveness of linear quadratic Gaussian (LQG) controller minimizing the rotor displacement and the amplifier current nearby the first bending critical speed. Furthermore, the comparison between the proportional-integral-differential {PID) controller with phase lead compensator and the LQG controller verifies the superiority of LQG controller in reducing the amplifier currents. [ABSTRACT FROM AUTHOR]

Since the mass of the rotor in high energy density magnetically suspended motor (HEDMSM) is always large and there are only three balancing planes on the flexible rotor restricted by the structure of the motor, which means that the second bending mode cannot be balanced using N + 1 planes method which is always applied to balance the flexible rotor. Then, the rotor displacements maybe large and this situation will make the system consume large amplifier currents when the rotor passes the first bending critical speed. Therefore, the mode separation method is proposed to separate the first and the second bending modes in rotor displacement and reconstruct the displacement signal nearby the first bending mode. Then, the original rotor displacement signal used by the digital controller is substituted by the reconstructed displacement signal and the amplifier current is reduced a lot when the rotor passes the first bending critical speed. Finally, the experiment of mode separation is carried out in 100 kW magnetically suspended motor and the experiment results show the effectiveness and superiority of the mode separation method in reducing the amplifier current when the rotor passes the first bending critical speed. [ABSTRACT FROM AUTHOR]

The rated rotational speed of the magnetically suspended motor (MSM) is always above the bending critical speed to achieve high energy density. The rotor will have a dramatic resonance when it passes the critical speed. Then, the magnetic bearing has to provide large bearing force to suppress the synchronous vibration. However, the bearing force is always limited by magnetic saturation and power amplifier voltage saturation. This paper proposed an optimum damping control method which can make effective use of the limited bearing force to minimize the synchronous vibration amplitude of the rotor nearby the critical speed. The accurate rotor model is obtained by theoretical analysis and system identification. The unbalance force response of the bending mode of the rotor is analyzed. The small gain theorem is used to determine the range of the magnitude of the control system. Then, the relationship of the optimum damping varying with the magnitude and phase of the control system nearby the critical speed is analyzed. The run-up experiments are carried out in 315 kW MSM and the results show the effectiveness and superiority of the optimum damping control method. [ABSTRACT FROM AUTHOR]

Investigating dynamics and stability of rotors supported on journal bearings is a crucial step in the design of an efficient and reliable rotating machine. In the current work, a model for flexible rotor supported on two symmetric journal bearings is investigated. The nonlinear bearing forces are evaluated by either using direct solution of Reynolds equation or analyzing Reynolds equation to obtain linear and nonlinear bearing stiffness and damping coefficients using time dependent second order perturbation method. These coefficients are obtained for different operating conditions and bearing parameters such as length to diameter ratio, groove angle or applied groove pressure. The present results are validated with the previous literature and a perturbation analysis is used to investigate the validity range of the bearing linear and nonlinear coefficients. A novel technique based on polynomial fitting is used to present the bearing coefficient as a function of the bearing parameters. This enables the investigation of the dynamics of flexible rotor model using numerical continuation technique. Also, the effect of the bearing design parameters such as groove angle, length to diameter ratio and static pressure on the system stability is investigated. • Nonlinear journal bearing coefficients are investigated. • Time dependent perturbation method is used to obtain bearing coefficients. • Parametric analysis to the journal bearing coefficients is introduced. • Continuation analysis to the dynamics of rotor supported on Journal bearing is introduced. [ABSTRACT FROM AUTHOR]

The industry application of magnetic bearing (MB) has recently extended from rigid rotor system to flexible rotor system; therefore, the design method of MB for flexible rotor is required. To reduce the inductance of the control coils and improve the dynamic performance, a radial MB with novel coil arrangements and connections is proposed in this paper. An optimal design method of the proposed MB is presented, which includes the simultaneous consideration of the MB itself, rotor-dynamics and the characteristics of the controller, and power amplifier (AMP). The equivalent magnetic circuit, which takes eddy current effects into considerations, is used as analytical model; the damping coefficient at resonant frequency is used as the objective function. The influence of the bias flux density on the objective function is analyzed, and the results show that increasing the bias flux density can reduce the burden on the AMP. Practical restrictions are also imposed, such as flux density constraint, maximum supply voltage of AMP, winding space constraint, pole width constraint, and copper wire capacity constraint on the design. By the optimization method of genetic algorithm and branch and bound algorithm, the optimal structure parameters of the proposed MB have been obtained. Finally, a design example is given and a prototype is manufactured. The experimental results demonstrate that the proposed MB and the optimization method are feasible and valid in the application to high-speed flexible rotor. [ABSTRACT FROM AUTHOR]

The high speed flexible coupling (HSFC) is used on the fighter aircraft gas turbine engines utilizing non-lubricated, metallic flexible plates. The HSFC has to transmit power at high speeds typically ranging from 10,000 to 18,000 rpm from engine gear box to accessory gear box. The HSFC also accommodates large axial and lateral misalignments due to thermal expansion and flight dynamics. The imposed angular and axial displacement distorts or bends the flexible plates. To achieve an infinite life design for HSFC, it is necessary to keep away from the different natural frequencies from the operating range of speeds apart from the optimal structural strength to accommodate larger misalignment. This paper discusses about details of system analysis, design and evaluation of HSFC for gas turbine engines. The dynamic characteristics of the HSFC are investigated through the finite element analysis (FEA) and experimental verification. The present approach can be extended for the development of similar type of flexible couplings. [ABSTRACT FROM AUTHOR]

Abstract: Active magnetic bearings (AMBs) are widely applied in high-speed rotating machinery, especially in special environments. In designing and adjusting an AMB system, the mathematical model of the system plays an important role. Identification is a useful method to obtain the models of AMB systems. This paper concentrates on identification method for an AMB system with a flexible rotor. Based on the theoretical system model and the measured frequency–response model, the proposed method estimates the unknown parameters and establishes the transfer function matrix model of the AMB system. According to the theoretical model, this paper decomposes the identification procedure into a few steps and the model is sequentially reduced by these steps. In this procedure, the submodels are identified separately and finally combined together. The proposed method is validated by experiments on three AMB systems. [Copyright &y& Elsevier]

A flywheel energy storage system (FESS) with a permanent magnet bearing (PMB) and a pair of hybrid ceramic ball bearings is developed. A flexibility design is established for the flywheel rotor system. The PMB is located at the top of the flywheel to apply axial attraction force on the flywheel rotor, reduce the load on the bottom rolling bearing, and decrease the friction power loss. The magnetic force of the PMB is analyzed through finite element method, and the force with the air gap of the PMB is verified experimentally. A squeeze film damper (SFD) is introduced to support the bottom rolling bearing, suppress lateral vibration, and enhance the stability of the flywheel rotor system. A dynamic model of FESS is established through transfer matrix method, Jones-Harris rolling bearing theory, and a finite length bearing dynamic model for SFD, which is verified by measuring the amplitude-frequency response. The effect of SFD radial clearance and unbalanced mass distribution on the dynamics of FESS is discussed. A spin-down test for the FESS prototype is conducted in a moderate vacuum. Results show that the hybrid bearing and flexibility design for the rotor system allow for the use of a small rolling bearing to reduce the power loss of FESS caused by friction. The developed FESS is simple in structure, stable without active control, low in cost, and convenient in maintenance. [ABSTRACT FROM AUTHOR]

For a magnetically levitated flexible rotor (MLFR), the amount of residual imbalance not only generates undesired vibrations, but also results in excessive bending, which may cause it hit to the auxiliary bearings. Thus, balancing below the critical speed is essential for the MLFR to prevent the impact. This paper proposes a balancing method of high precision and high efficiency, basing on virtual trial-weights. First, to reduce the computed error of rotor's mode shapes, a synchronous notch filter is inserted into the active magnetic bearing (AMB) controller, achieving a free support status. Then, AMBs provide the rotor with the synchronous electromagnetic forces (SEFs) to simulate the trial-weights. The SEFs with the initial angles varying from 0 deg to 360 deg in the rotational frame system result in continuous changes in the MLFR's deflection. Last, correction masses are calculated according to the changes. Compared to the trail-weights method, the new method needs not test-runs, which improves the balancing efficiency. Compared to the no trail-weights method, the new method does not require a precise model of the rotor-bearing system, which is difficult to acquire in the real system. Experiment results show that the novel method can reduce the residual imbalance effectively and accurately. [ABSTRACT FROM AUTHOR]

[Display omitted] • Mathematical modelling of misaligned sensors and flexible rotor-AMB system. • Dynamic effects of AMBs and sensors residual misalignments. • Innovative virtual trial misalignment (VTM) approach. • Identification of unbalance, sensors and AMBs residual misalignment. • Estimation of AMBs force-displacement and force-current stiffness parameters. A model-based identification algorithm derived from an innovative virtual trial misalignment (VTM) strategy is developed to identify the rotor unbalance, active magnetic bearings (AMBs) residual misalignment and their displacement and current stiffness parameters. The VTM approach is aligned with the similar concept of trial unbalance utilized in balancing of the rotor. This virtual method for providing the trial misalignment to the rotor is more reliable and effective than the physical trial misalignment, as the method does not need test runs and a precise model of the rotor-sensor-bearing system. The mathematical modelling for the non-contact displacement sensors offset from the rotor center has also been presented and even the offset amounts are identified using this novel approach. To illustrate the proposed methodology, an unbalanced and misaligned flexible rotor system mounted on multiple AMBs has been considered. Non-collinearity between the rotor operating axis and the axis of supported AMBs is the prime cause of misalignment (combination of parallel and angular) fault. A finite element method with a gyroscopic dynamic condensation scheme is employed for acquiring the reduced form of system equations. The numerically generated frequency domain responses are further employed in the identification algorithm for estimating the rotor-AMB system and faults parameters. Besides this, the different levels of the random noise signal as well as rotor modelling errors are added in the developed VTM based methodology to check its efficacy and sensitiveness in a practical rotor model. [ABSTRACT FROM AUTHOR]

• Mechanism of energy conversion efficiency improvement with flexible rotors studied. • 3D finite element model with coupling among various parameters established. • Transient solution and modal analysis methods applied on TRUM-60. • Results and validated experimentally as well as using 3D vibration measurements. Flexible rotors are widely used in traveling wave rotary ultrasonic motors (TRUMs) because of their higher energy conversion efficiency; however, there have been few reports on how flexible rotors improve the energy conversion efficiency of ultrasonic motors. In this study, we investigate the improvement mechanism of energy conversion efficiency in TRUMs with flexible rotors. A 3D finite element (FE) model with full coupling among a piezoelectric coupled stator, rotor, friction layer, and the rigid–elastic contact interface of the stator and friction layer is established. To analyze the mechanism by which the efficiency of the TRUM is improved, the contact interface and rotor vibration information are extracted. Taking TRUM-60 as an example, the transient solution method and modal analysis method are used to solve the model. It is found that when the stator mode is B 09 , the flexible rotor mode is B 19. The energy conversion efficiency of the TRUM is obtained from the ratio of output power to the electrical input power of the model solution. The results are validated using 3D vibration measurements and energy conversion efficiency experiments. The simulation result shows that the motor with flexible rotor improves the energy conversion efficiency compared with the motor with rigid rotor, which can be attributed to two reasons: first, the axial amplitude ratio of the flexible rotor to the stator is reduced; second, the flexible rotor reduces the radial friction. This study reveals the influence of flexible rotor on the output efficiency and can thus provide guidance for rotor design. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents experimental optimization of a hybrid foil–magnetic bearing to support a flexible rotor. Hybrid foil–magnetic bearing is a synergistic cross-fertilization of two oil-free bearing technologies: air–foil and magnetic bearing. First, an experimental set-up for a flexible rotor supported by a hybrid foil–magnetic bearing is built. Then, unbalance responses of the flexible rotor supported by two different bearings: air–foil and hybrid bearings are compared. The unbalance responses show that the rotor vibration in bending critical speed is more suppressed by the hybrid foil–magnetic bearing rather than the air–foil bearing only. Experimental optimization of the hybrid foil–magnetic bearing is performed by tuning control gains and offsets of the magnetic bearing. The resulting optimized controller for the hybrid foil bearing would not only improve vibration performance by 26% at bending critical speed, but also save the energy consumption of driving motor by 50% during run up test. [Copyright &y& Elsevier]

Highlights: [•] Flexible rotor balancing with active magnetic bearings. [•] Experimental estimation of active magnetic bearings dynamic parameters. [•] Investigation of measurement noise and modelling error. [ABSTRACT FROM AUTHOR]

Synchronous lateral vibration is a frequent cause of machine failure and is probably the most common source of machine noise and vibration. Among the broad ranges of faults that cause synchronous vibration, mass unbalance and bow are the most common faults present in the rotor and hence their detailed understanding is required. In the present study, a rotor model having both mass unbalance and bow is analyzed to find the influence of these faults on the synchronous response. The synchronous response of a rotor is a function of influence coefficients due to mass unbalance and bow. The influence coefficients are analytically derived using transfer matrix method for rotor bearing system having both mass unbalance and bow. The correction unbalance vector required to balance the rotor at its first bending critical speed is computed using these analytically derived influence coefficients and the rotor responses measured at a speed much lower than the first bending critical speed. The balancing method needs a good mathematical model for the rotor system in finding the influence coefficients for mass unbalance and bow. This procedure helps in estimating the correction mass in a single trial run and using a single balancing plane, thus reducing the cycle time for balancing flexible rotors. Experiments are carried out on a test rig to verify the procedure. [ABSTRACT FROM AUTHOR]

The majority of applications of active magnetic bearings were still based on conventional PID control system. The 2 DOF (two-degree-of-freedom) PID controller can solve the problem of the optimal tuning for the disturbance response and the one for the set-point response. A 2 DOF PID control for active magnetic bearings (AMB) is discussed in this paper. According to fundamental principle of finite element mathematical model of the flexible shaft is presented. The controlled plant of 2 DOF PID mathematical model is determined. Control parameters of the 2 DOF PID systems for active magnetic bearings are discussed. The active magnetic bearing laboratory system is shown. The obtained experimental results show that 2 DOF PID controller of the active magnetic bearings-flexible rotor system is stable through the first critical speed. [ABSTRACT FROM AUTHOR]

This paper describes efforts required to operate a cogeneration facility's extraction-condensing steam turbine generators with flexible rotor design on a slow-roll mode of operation (also called true stand-by). Slow-roll mode of operation became necessary due to changing steam demand from the host refinery, resulting in decreased performance of the steam turbine generators and their associated economic losses. Design modifications implemented to achieve slow-roll of steam turbine generators without affecting reliability and availability of the entire cogeneration facility are discussed in this paper. Successful implementation of the design modifications was demonstrated via extensive field testing on two steam turbine generator units during the summer of 2011. A simple life cycle economic analysis shows the payback period for the project is approximately seven months. [ABSTRACT FROM AUTHOR]

With the continuous improvement of aviation engine performance, the engine rotor’s working speed is above the first critical speed, and some small engine rotor’s operating speed exceeds the second critical speed, thus most of the aircraft engine rotor is now flexible rotor. The primary goal of this research is to provide a new method without trial weights for flexible rotor balancing, which is based on the traditional modal balancing method and combines the dynamic characteristics of the rotor, such as the model shape, obtained by using finite element analysis software, the critical speed, the modal damping ratio, and so on. To demonstrate the validity and accuracy of the new method, two experiments are carried out. One is on a four-disk simply supported structure rotor experiment, which is balanced by using three balancing planes. After balancing the amplitude of the rotor through the first critical speed is reduced 80%. The other is on a turboshaft engine power turbine rotor which had been balanced quite well. The initial amplitude of the turboshaft engine power turbine rotor is 0.05 mm when it rotates at low speed, and the maximum amplitude is just 0.12 mm when it rotates at the first critical speed. After balancing the amplitude of the rotor through the first critical speed is reduced to 0.06 mm. The results of the experiments indicate that the new without trial weights method is a significant effective method for the flexible rotor dynamic balancing, and it will be widely used to balance the engine rotor. [ABSTRACT FROM PUBLISHER]

Nearly a century ago, the liquid self-balancing device was first introduced by M. LeBlanc for passive balancing of turbine rotors. Although of common use in many types or rotating machines nowadays, little information is available on the unbalance response and stability characteristics of this device. Experimental fluid flow visualization evidences that radial and traverse circulatory waves arise due to the interaction of the fluid backward rotation and the baffle boards within the self-balancer annular cavity. The otherwise destabilizing force induced by trapped fluids in hollow rotors, becomes a stabilizing mechanism when the cavity is equipped with adequate baffle boards. Further experiments using Particle Image Velocimetry (PIV) enable to assess the active fluid mass fraction to be one-third of the total fluid mass. An analytical model is introduced to study the effects of the active fluid mass fraction on a flexible rotor supported by flexible supports excited by bwo different destabilizing mechanisms; rotor internal friction damping and aerodynamic cross-coupling. It is found that the fluid radial and traverse forces contribute to the balancing action and to improve the rotor stability, respectively. [ABSTRACT FROM AUTHOR]

Abstract: Flexible rotor is a crucial mechanical component of a diverse range of rotating machineries and its condition monitoring and fault diagnosis are of particular importance to the modern industry. In this paper, Bayesian belief network (BBN) is applied to the fault inference for rotating flexible rotors with attempt to enhance the reasoning capacity under conditions of uncertainty. A generalized three-layer configuration of BBN for the fault inference of rotating machinery is developed by fully incorporating human experts’ knowledge, machine faults and fault symptoms as well as machine running conditions. Compared with the Naive diagnosis network, the proposed topological structure of causalities takes account of more practical and complete diagnostic information in fault diagnosis. The network tallies well with the practical thinking of field experts in the whole processes of machine fault diagnosis. The applications of the proposed BBN network in the uncertainty inference of rotating flexible rotors show good agreements with our knowledge and practical experience of diagnosis. [Copyright &y& Elsevier]

In this paper the procedure of vibration suppression of a flexible rotor supported by active magnetic bearings (AMB) is presented. The purpose of the research is to establish stable rotor operation while passing the first critical speed. In order to create an efficient control algorithm, initially the model is reduced using the modal decomposition method. Afterwards, the reduced finite element model of the rotor is imported into the Matlab environment, where the AMB system control is designed by applying extensive rotordynamic synthesis, eigenvalue, and mode shape analysis as well as frequency and time response studies. The rotor system model considers the AMB sensor-actuator non-collocation. The results of this research show that minimal correction in the AMB position and orientation could provide a great improvement in the AMB tunability and overall system stability. [ABSTRACT FROM AUTHOR]

Abstract: The nonlinear dynamic behavior of a rotor–bearing system with interaction between blades and rotor is addressed in this paper. Using the Lagrange equation, a time-dependent nonlinear model of a flexible blade–rotor–bearing system is established, in which the rotor is supported by journal bearings and the blades are modeled as pendulums in order to analyze the dynamic coupling between the elastic blades and the flexible shaft. To emphasize the gyroscopic effect of the rotor, the disk is assumed to be located at an arbitrary position of the shaft. Employing the orthogonal transformations, the 1-nodal diameter equations of motion of the blades, which are coupled with the dynamic equations of the rotor, are decoupled with other equations of the blades. Then the parametric excitation terms in the blade–rotor–bearing system are simplified in terms of periodic transformations. The dynamic equations with nonlinear oil-film forces are numerically solved using the Runge–Kutta method. Bifurcation diagrams, three-dimension spectral plots, and Poincaré maps are employed to analyze the dynamic behavior of the system. The numerical results show that the nonlinear dynamic behavior of the system varies with the increase of the rotational speed. And the effect of the nonlinear vibration of the rotor on the blade vibration is discussed. [Copyright &y& Elsevier]

Abstract: The Hermitian matrix of influence coefficients is one indicator of a balancing experiment’s overall effectiveness. The condition number of this matrix is determined by the sensor locations and balancing planes selected for the balancing procedure. In this research, we use finite element analysis to simulate the balancing of flexible rotor-bearing systems under various arrangements of sensors and planes. The condition number turns out to be inversely related to the accuracy of the balancing and directly related to the sum of correction masses; in other words, the accuracy and efficiency of rotor balancing can be improved by selecting sensor locations and balancing planes which reduce the condition number. This result is validated by experimental measurements of a physical rotor system. [Copyright &y& Elsevier]

Abstract: Most high-speed rotating machineries operate at a speed higher than their first critical speeds. Since there are two different modal shapes existing in rotors under and over that speed, the most currently used balancing techniques for rotors require two steps to fulfill the balancing purpose. Firstly, a traditional low-speed rigid balancing is performed, and secondly, a high-speed balancing procedure is carried out. This high-speed balancing procedure is very complicated and dangerous; it needs large number of test runs and causes corresponding wastages as well. This paper develops a novel balancing method, called Low-Speed Holo-Balancing (LSHB), which can balance a flexible rotor without test runs at high speeds. The principle of LSHB is mainly based on the holospectrum technique, which will be explained in detail at the end of this paper. Instead of using information from a single sensor, through a multi-sensor fusion a three-dimensional holospectrum is constructed to describe the vibration response of a rotor vividly and accurately. LSHB can easily balance the flexible rotors at lower speeds than the first critical speed, according to the decomposition results of holospectrum. Because LSHB does not need any test runs at or higher than the first critical speed like any other traditional balancing techniques do, the balancing procedure is much safer, and economical. The key points and main operating steps of LSHB will be presented. The experimental results conducted on a flexible-rotor test rig validated the effectiveness of this method. [Copyright &y& Elsevier]

Abstract: This study deals with active robust modal vibration control of rotor systems supported by magnetic bearings. The inherent divergent rigid body modes are suppressed by using a dual-level control approach. Finite element method is applied to formulate the rotor model. The Timoshenko beam theory, including the effects of shearing deformations and rotary inertia, is considered in this work. Because practical control systems are often limited by its sensing, hardware, and computation speed capabilities, the reduced order approach is often used for a control system design. This study applies the independent modal space control (IMSC) approach to extract the accurate lower modes from the complex rotor systems with the gyroscopic effect considered. In practice, it is extremely difficult to model the complete dynamic characteristics of a rotor system. The model may contain unmodelled dynamics and parameter changes, which can be viewed as uncertainties of a system. As opposed to the conventional control approach, which requires fixed and accurate system parameters, this study considers robust control approach to design a controller capable of tolerating external disturbance and model uncertainties. It is demonstrated that the proposed approach is effective for vibration suppression when the system is subjected to impulsive or step loading, speed variation, and sudden loss of disk mass. [Copyright &y& Elsevier]

Abstract: The motion equations have been established for symmetrical single-disk flexible rotor-bearing system, and non-linear oil-film forces of finite journal bearings are calculated. The rotor''s stiffness and damping are considered. The motions of journal and disk have been simulated with fourth-rank Runge-kutta method. The threshold speed of the system based on linear oil-film forces has been derived. Non-linear transient simulation and unbalanced responses are investigated. Moreover, linear oil-film forces are used for computing unbalanced responses, and the feasibility is discussed. [Copyright &y& Elsevier]

In this study, a moderately flexible rotor is investigated experimentally in a water channel. The rotor consists of a single blade with a simplified rectangular geometry and is tested with various pitch angles at different rotational speeds. The flapwise bending deformation and induced torsion of the blade are measured using image processing tools, and flow fields are extracted with Particle Image Velocimetry. The blade deformation behaviour depends strongly on the pitch angle, it includes extreme downstream and upstream bending for high negative and positive pitch, respectively. In addition, large-amplitude, low-frequency bending fluctuations are observed in a certain range of rotational speeds for negative pitch. The wake vorticity fields show large-scale recirculation zones being formed and intermittently shed behind the rotor, resembling the dynamics of the Vortex Ring State known from helicopter aerodynamics. A comparison with the flow generated by a rigid rotor of the same geometry is also carried out. [Display omitted] [ABSTRACT FROM AUTHOR]

• Experimental investigation of the Morton effect on a simple rotordynamic test rig. • Journal transient temperatures are presented. • The rotor operates close to a flexible mode. • The results underline the importance of the mechanical unbalance. • Spiral synchronous vibrations are highlighted. The Morton effect is a thermally induced increase of synchronous vibrations amplitudes of rotors. The shearing of the fluid film in a journal bearing produces an asymmetric heating of the rotor which causes its thermal bending and increases the unbalance. The vector of synchronous vibrations describes a spiral with slowly increasing amplitude and phase change. The thermal bending is also enlightened by a hysteresis effect of the synchronous vibrations during start-up and coast-down. The purpose of this study is to highlight experimentally the signature of Morton effect in a flexible rotor supported by a flexure pivot tilting pad journal bearing. A dedicated test rig has been built. It consists of a flexible rotor guided at one end by a ball bearing and at the other end by a flexure pivot tilting pad bearing. The tests consist of controlled start-ups to a speed close to the critical speed of the first rotor flexible mode, followed by controlled coast-downs. They show a hysteresis of the amplitudes and phases of the synchronous vibrations. Other tests performed at constant speeds show amplitude and phase variations corresponding to spiral vibrations. The results are explained by the non-uniform distribution of the rotor temperature. [ABSTRACT FROM AUTHOR]

• High-speed balancing of the flexible rotor system integrated with AMBs. • Application of virtual trial unbalances as magnetic excitation forces. • Force current stiffness factor estimation of the AMBs. • Utilization of half-spectrum amplitudes and phases of ramp-up responses. • Repetitive flexible rotor balancing using unbalance estimates at low-speeds. An identification algorithm is developed to execute high-speed balancing of the flexible rotor system supported on the conventional bearings, while rotating the system below its critical speed. The system is integrated with Active Magnetic Bearings (AMBs) as a suppression actuator as well as an excitation actuator. The AMB suppresses the vibration of the system and allows to introduce virtual trial unbalances to identify the orientation and magnitude of the residual unbalances present in the system. An Advanced Influence Coefficient Method (AICM) is developed that utilizes the influence coefficients obtained at high speed and unbalances identified at the low speed to effectively estimate the balance masses required for the high-speed flexible rotor balancing. The influence coefficients are required to be obtained just the once, whereas the balance masses can be estimated periodically by identifying the unbalances at low speeds. After balancing, the system can easily cross its critical speeds with less vibration. The AMB also controls abrupt change in vibration amplitude due to any uncertain faults, while operating at high speeds. Additionally, the virtual trial unbalances as a magnetic force generated through AMB reduces the mechanical effort involved in the placing of physical trial unbalances in the rotor system. This method allows on-site condition monitoring of the system periodically to reduce the impairment of high-speed machinery. Image, graphical abstract [ABSTRACT FROM AUTHOR]

The paper describes a method to perform the dynamic balancing of multiple speeds flexible rotors. The method is integrated with a dual-objective optimization method (DOM) to overcome the shortcomings that classical least squares influence coefficient method (LSM) may induce by generating excessive residual vibrations at specific rotor speeds when a flexible rotor is dynamically balanced. The technique here described involves two consecutive optimization steps. During the first step an optimal correction weight of the parameters population is identified by using a genetic algorithm (GA) applied to an objective function made from the sum of the squares of the residual vibrations. This first optimization step is performed multiple times to obtain a set of optimized correction weights. A second optimization process is then performed by minimizing the maximum value of the residual vibration of the rotor. Each correction weight belonging to the first optimization set is embedded into an equation that represents the residual vibration to obtain the final optimal correction weight. The validity of the proposed method is verified by simulations and experiments related to a double-disc rotor-bearing system. The results show the DOM provides a more effective balancing strategy than the one obtained from using GA and classical LSM. • Balancing method of flexible rotors combined with a dual-objective optimization. • Global search and multi-point optimization developed for flexible rotor balancing. • Method minimizes the square sum and the maximum value of the residual vibration. • The balancing method combined with dual-objective optimization is easy to implement. [ABSTRACT FROM AUTHOR]