Bending Critical Speed Research Articles

Development of Flexible Rotor Balancing Procedure Using Response Matching Technique

Abstract Purpose The two main methods under use for balancing of flexible rotor know as Influence Coefficient Method (ICM) and Modal Balancing Method (MBM). It needs few trials to arrive at predicting the unbalance mass. In presence of initial bow along with unbalance these methods will fail to give accurate predictions. It becomes further complex when rotors are mounted on flexible supports. To overcome the limitations of these two methods, a new method called response matching method is disused here for flexible rotor balancing. Method The objective of present work is to develop a balancing procedure for flexible rotor balancing, wherein unbalance response of Finite element model (FEM) is matched with experimental response using iterative technique. Unbalance masses, moments and shaft stiffness of FEM model was iterated to match experimental unbalance responses at three distinct speeds below its first flexural critical speed. Further, unbalances mass was estimated iteratively to cross the bending critical speed with lower residual amplitudes at critical speeds. Results Experimental test setup was developed to validate the proposed method. Results show that present method predicts the unbalance masses and moments to compensate the bow more accurately. Conclusion Results shows that with single run, rotor unbalance masses and movements are predicted accurately. Corrected masses are used in experimental test setup to check the rotor amplitudes during its bending critical speeds. It shows rotor passes first bending critical speed without excessive amplitudes.

Dynamic Response Analysis of a Magnetically Suspended Dual-Rotor System Considering the Uncertainty of Interference-Fit Value

Interference fit is often used in rotating machinery to transmit torque and force. The actual interference value is uncertain due to factors such as manufacturing errors and operating conditions, resulting in a gap between the response of the system and theoretical results. Therefore, the interval method is used to study the magnetically suspended dual-rotor system (MSDS) with uncertainty of interference-fit value. Firstly, a theoretical model of the MSDS was established using the finite element method, and the influence mechanism of the interference value on the rotor bending stiffness was derived. Then, the rotor stiffness range was obtained from the uncertain range of interference value. Finally, the dynamic response of the MSDS was studied based on the Chebyshev interval method. The research results indicate that the uncertainty of interference value has an effect on the vibration response of the MSDS. The vibration response of the system is most affected near the first-order bending critical speed, and the effect on rotor response is relatively small in other angular speed regions. The research results can provide a basis for the design of rotor systems.

Research on the Influence of Rub-impact Buffer Stroke on the Dynamic Response of Two-section Rotor

The two-section rotor needs to go through multiple critical speeds in the process of speed increase. When the rotor approaches the critical speed, its amplitude increases rapidly, easily causing collision damage. The auxiliary over-critical device cushions and protects the rotor through rubbing, which makes it pass the critical speed under the safe amplitude. In this paper, the influence of buffer stroke, the key structural parameter of the auxiliary over-critical device, on the dynamic response of the two-section rotor at the first-order bending critical speed is analyzed, the influence of its buffer stroke on the over-critical dynamic response of the rotor is calculated and analyzed, and the impact and friction force on the rotor is calculated. The design principles and results of the buffer stroke parameters of the auxiliary supercritical device are proposed.

Kriging-Based Surrogate Controller for Robust Control of a Flexible Rotor Supported by Active Magnetic Bearings

Abstract 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.

Multidisciplinary Design Overview and Comparison for Two Case Studies of High Speed Permanent Magnet Machines

High-speed electrical machines have become more and more popular applications driven by electric vehicles and all-electric aircraft. Their design usually requires a high level of integration, including loss, thermal, cooling, and mechanical aspects. This article makes a comprehensive overview of the multidiscipline design method of high-speed permanent magnet (HSPM) machines. A 4-kW machine with the flexible rotor operating above the first bending critical speed is presented, and a 30-kW machine is also developed with consideration of the multidisciplinary design process for contrast. The constraint and the mechanism in the overview are investigated to clarify the influence of different design variables on the rotor structure and performance. Finally, two machines are fabricated and tested, and the values are compared with theoretical results.

Critical Vibration and Control of the Maglev High-Speed Motor Based on μ-Synthesis Control.

The Maglev motor has the characteristics of high-speed and high-power density, and is widely used in compressors, molecular pumps and other high-speed rotating machinery. With the requirements of miniaturization and high speed of rotating machinery, the rotor of the maglev motor will operate above the bending critical speed, and the critical vibration control of the flexible rotor is facing challenges. In order to solve the problem of the critical vibration suppression of the maglev high-speed motor, the system model of the maglev motor is established, the rotordynamics of the flexible rotor are analyzed and the rotor model is modal truncated to reduce the order. Then, the μ-controller is designed, and the weighting functions are designed to deal with the modal uncertainty. Finally, an experimental platform of the maglev motor with the flexible rotor is built to verify the effect of the μ-control on the suppression of the critical vibration of the maglev rotor.

Development of a rotor test rig with a novel controllable preload foil-air bearing

The introduction of a lobed clearance profile into a foil-air bearing can improve rotordynamic performance through the delay of the onset of instability speed and consequent suppression of sub-synchronous vibrations. This can be achieved by shimming (preloading) the foil at three angular locations along its interface with the sleeve. This paper presents the design, manufacture and testing of a rotor test rig with a novel controllable preload foil-air bearing design. The design's motorised micrometer screw assemblies securely prescribe a given preload in real time, eliminating the introduction of potential problems associated with the structural compliance of piezoelectric preloading mechanisms. The shaft of a previous test rig is redesigned to achieve a significantly higher speed with the same motor. The design work is guided by simulations. Experiments confirm that the projected preload is sufficient to virtually eliminate sub-synchronous vibration over the operating speed range. The experiments also confirm that the top speed could be significantly increased by redesigning the shaft for a higher bending critical speed. Although preload increases the friction torque, this is mitigated through the application of diamond-like carbon to the journal, and can be further mitigated by applying preload only in regimes of sub-synchronous vibration activity.

Vibration Control for the Flexible Rotor with Piezoelectric Bearings Based on the Mixed Sensitivity Robust Controller

Active control of flexible rotors is a challenging issue in modern industries. This paper focuses on the synthesis of a mixed sensitivity robust controller for a linear parameter-varying (LPV) system. The objective is to control rotor vibration, especially when the rotor is passing the first two bending critical speeds. In the formulation of the problem for the controller, weighting functions are proposed based on the relationship between the desired shape of the open-loop transfer function and sensitivity functions of the closed-loop system. Recent research has highlighted the efficiency of mixed sensitivity robust controllers in stabilizing a wide range of magnetic bearing systems. Here, the method is extended to control the vibration of a piezoelectric bearing system. The experimental rotor features two unbalance-exited resonances within its operating range. Experimental results demonstrate good performance of the vibration reduction and the effectiveness of the design method.

Design of Ultra-High-Speed Motor for FCEV Air Compressor Considering Mechanical Properties of Rotor Materials

This paper proposes the design process of an ultra-high-speed surface-mounted permanent magnet synchronous motor for a fuel-cell electric vehicle air compressor. Proposed design process enables ultra-high-speed motor design while considering mechanical stresses of the rotor materials according to the temperature. In the rotor design stage, the worst temperature condition of the permanent magnet and retaining sleeve on mechanical stress is investigated and reflected using analytical method. Rotor dimension such as permanent magnet and retaining sleeve thickness is determined considering both electromagnetic performance and mechanical characteristics. Then, from the initially designed model, the eddy current loss according to the shape ratio (SR) and torque density (TD) is analytically derived using proportional equation and finite-element analysis result. Then, SR and TD to reduce the eddy current loss and maximize the efficiency are determined while considering the first bending critical speed. Finally, to verify the proposed design process, the prototype is fabricated and tested. Test results show good agreement with the finite-element analysis results, and it is confirmed that the rotor is operated at the highest rotational speed without mechanical failure.

Simple Contact Stiffness Model Validation for Tie Bolt Rotor Design With Butt Joints and Pilot Fits

Abstract Tie bolt rotors for centrifugal compressors comprise multiple shaft components that are held together by a single tie bolt. The axial connections of these rotors—including butt joints, Hirth couplings, and Curvic couplings—exhibit a contact stiffness effect, which tends to lower the shaft bending frequencies compared to geometrically identical monolithic shafts. If not accounted for in the design stage, shaft bending critical speed margins can be compromised after a rotor is built. A previous paper had investigated the effect of tie bolt force on the bending stiffness of stacked rotor assemblies with butt joint interfaces, both with and without pilot fits. This previous work derived an empirical contact stiffness model and developed a practical finite element modeling approach for simulating the axial contact surfaces, which was validated by predicting natural frequencies for several test rotor configurations. The present work built on these previous results by implementing the same contact stiffness modeling approach on a real tie bolt rotor system designed for a high pressure centrifugal compressor application. Each joint location included two axial contact faces, with contact pressures up to five times higher than previously modeled, and a locating pilot fit. The free-free natural frequencies for different amounts of tie bolt preload force were measured, and the frequencies exhibited the expected stiffening behavior with increasing preload. However, a discontinuity in the data trend indicated a step-change increase in the contact stiffness. It was shown that this was likely due to one or more of the contact faces becoming fully engaged only after sufficient tie bolt force was applied. Finally, a design calculation was presented that can be used to estimate whether contact stiffness effects may be ignored, which could simplify rotor analyses if adequate contact pressure is used.

Experimental and Analytical Investigation of Short Squeeze-Film Damper (SFD) Under Circular-Centered Orbit (CCO) Motion

Modern rotating machines operate at higher shaft speeds, crossing a few bending critical speeds, which often induce large dynamic loads on the bearing supports that may increase the rotor orbital whirl motion. Accurate prediction of squeeze-film damper (SFD) forces is the key issue for designers to arrive at a suitable design. An experimental setup was developed with eccentric shaft to find the damper forces. Submerged type damper with relatively large clearance was considered for the present work. It results in higher gap Reynolds number (Re) which varies from 1.5 to 15 in the present case. Four force sensors are used to measure the damper forces and two eddy current probes were used to measure the damper orbit. Circular-centered orbit (CCO) of eccentricity ratio (e) 0.176 was used in the present work. Theoretical modeling of SFD forces is considered with viscous, inertial, temporal contributions under laminar and turbulent conditions. Modified Reynolds equation with short damper approximation is used to derive the SFD forces for 2π- film. Fourier coefficients of measured forces and displacements are estimated and used to extract the 1× components and phase information. Radial and tangential forces were calculated from the measured total forces to find the contributions of viscous and inertial forces. Experimental forces are compared with theoretically predicted forces and they are in good agreement with the experimental results.

Design of active magnetic bearing rotors and their control method for passing through fourth bending critical speed

Design of active magnetic bearing rotors and their control method for passing through fourth bending critical speed

Active vibration control of the flexible high-speed rotor with magnetic bearings via phase compensation to pass critical speed

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.

10 MW급 초임계 CO₂ 발전용 축류 터빈의 구성 설계 및 회전체 동역학 해석

This paper presents component design and rotordynamic analysis results of axial turbine applied to a 10 MW super-critical CO₂ cycle. The axial turbine consists of 3 stage turbine blades and a shaft supported by two fluid film bearings. To prevent the leakage, we installed two seals near the turbine inlet and exit side. We adopts a tilting pad bearing for the stable operation of the rotor. The tilting pad bearing consists of 5 pads and leading edge grooves are installed for the effective thermal management. Based on the rotor layout, we conduct a rotordynamic analysis. The dynamic coefficients of the tilting pad bearings were calculated based on the thin film lubrication theory. Turbine blades, thrust collar and seals were modeled as equivalent inertia. The predicted Campbell diagram showed that there are three critical speeds, namely the conical and the first , second and third bending critical speeds under the rated speed and it is predicted that the rotor have 20% separation margin for the fourth bending critical speed. In addition, unstable modes are not presented under the rated speed and the unbalance response prediction showed that vibration levels are controlled within 0.02 mm for all speed ranges owing to the high damping ratio of the modes.

Design, modeling, and robust control of the flexible rotor to pass the first bending critical speed with active magnetic bearing

Recently, magnetic bearings have been applied to many rotating machines such as turbo-molecular pumps, cooling gas compressor, flywheel energy storage systems. And high-power density is the future development trend of these machines, which demands that the rotor characterizes slender and high rotating speed and operates above the critical speeds. However, it is a big challenge for a flexible rotor to pass the bending critical speeds and operate above the critical speeds steadily and reliably. Based on above reasons, this article presented the design, modeling, and analysis framework of a flexible rotor test rig with active magnetic bearing (AMB). Special attention was paid to the flexible rotor dynamic model development, dynamic analysis, and model-based robust control design. First, the main structure features were illustrated in detail, including the key dimensions and parameters of the AMB components and rotor. Then models of components including power amplifier and displacement sensor were developed. The finite element method based on Timoshenko beam theory was applied to the rotor dynamics model. The dynamic analysis of the rotor is the foundation of the controller design. Hence, modal analysis had been done, obtaining rotor dynamic properties via synthesis of natural frequency, mode shapes, and Campbell plots. Reduced order model was obtained for controller design convenience. A robust H∞ controller was designed based on the system model and controller performance was validated with numerical simulation. The results indicate that the controller has good vibration suppression performance and makes the flexible rotor pass the first bending critical speed.

Identification of dynamic parameters of active magnetic bearings in a flexible rotor system considering residual unbalances

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.

Investigations on bending-torsional vibrations of rotor during rotor-stator rub using Lagrange multiplier method

The ever-increasing need of highly efficient rotating machinery causes reduction in the clearance between rotating and non-rotating parts and increase in the chances of interaction between these parts. The rotor-stator contact, known as rub, has always been recognized as one of the potential causes of rotor system malfunctions and a source of secondary failures. It is one of few causes that influence both lateral and torsional vibrations. In this paper, the rotor stator interaction phenomenon is investigated in the finite element framework using Lagrange multiplier based contact mechanics approach. The stator is modelled as a beam that can respond to axial penetration and lateral friction force during the contact with the rotor. It ensures dynamic stator contact boundary and more realistic contact conditions in contrast to most of the earlier approaches. The rotor bending-torsional mode coupling during contact is considered and the vibration response in bending and torsion are analysed. The effect of parameters such as clearance, friction coefficient and stator stiffness are studied at various operating speeds and it has been found that certain parameter values generate peculiar rub related features. Presence of sub-harmonics in the lateral vibration frequency spectra are prominently observed when the rotor operates near the integer multiple of its lateral critical speed. The spectrum cascade of torsional vibration shows the presence of bending critical speed along with the larger amplitudes of frequencies close to torsional natural frequency of the rotor. When m×1nX frequency component of rotational frequency comes closer to the torsional natural frequency, stronger torsional vibration amplitude is noticed in the spectrum cascade. The combined information from the stator vibration and rotor lateral-torsional vibration spectral features is proposed for robust rub identification.

Nonlinear responses analysis caused by slant crack in a rotor-bearing system

A finite element model of a slant cracked rotor system attached with two disks is presented. A slant crack model is adopted to simulate time-varying stiffness caused by shaft crack. Two types of bearing force (linear and nonlinear bearing forces) are used for simulating the bearing. This study focuses on the effects of eccentric phase differences of two disks on the nonlinear responses of the rotor-bearing system under steady-state process (constant rotating speed) and run-up process. The results show that for the lateral vibration, the superharmonic resonance phenomenon related to the first bending critical speed can be observed under linear bearing forces; however, it is almost unseeable under nonlinear bearing forces. For the torsional vibration, the superharmonic resonance phenomena related to the first torsional natural frequency appear under linear and nonlinear bearing forces. Large eccentric phase differences of two disks can decrease the rotor vibration and restrain the oil-film instability, and the angular acceleration can restrain the oil-film instability due to the tangential inertia force. Moreover, the large torsional amplitude of the second harmonic frequency can also be identified as a typical feature during run-up.

Identification of the dynamic parameters of active magnetic bearings based on the transfer matrix model updating method

The stiffness and damping coefficients of Active magnetic bearings (AMBs) have a great impact on the dynamics of a high-speed rotor AMB system, from its bending critical speed to the modes of its vibration and stability. To accurately obtain the stiffness and damping coefficients of AMBs, this study proposes a new identification approach based on the transfer matrix model updating method. By minimizing the error between the unbalance response calculated through the transfer matrix approach and the experimental measurements, the stiffness and damping coefficients are obtained using the simplex optimization algorithm based on the updating method of the model. According to the experimental data, we identify the parameters from 20 Hz to 260 Hz (1200 rpm to 15600 rpm). To verify the identified results, a finite element rotor AMBs model is created, and the theoretical unbalance response is predicted using the identified parameters. The theoretical unbalance responses closely coincide with the experimental measurements, indicating the effectiveness of the proposed method.

Active magnetic bearings dynamic parameters identification from experimental rotor unbalance response

Active magnetic bearings (AMBs) support rotors using electromagnetic force rather than mechanical forces. It is necessary to accurately identify the AMBs force coefficients since they play a critical role in the rotordynamic analysis including system stability, bending critical speeds and modes of vibrations.This paper proposes a rotor unbalance response based approach to identifying the AMBs stiffness and damping coefficients during rotation. First, a Timoshenko beam finite element (FE) rotor model is created. Second, an identification procedure based on the FE model is proposed. Then based on the experimental rotor unbalance response data from 1200rpm to 30,000rpm, the AMBs dynamic force parameters (stiffness and damping) are obtained. Finally, the identified results are verified by comparing the estimated and experimental rotor unbalance responses, which shows high accuracy.