Zeitschriftenartikel: „Flexible rotor“

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Barzdaitis, Vytautas, Vytautas Žemaitis, R. Jonušas, Vytautas Kazimieras Augustaitis und Vytautas Bučinskas. „Dynamics of a Mechatronic System with Flexible Vertical Rotor“. Solid State Phenomena 113 (Juni 2006): 223–28. http://dx.doi.org/10.4028/www.scientific.net/ssp.113.223.

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The paper is dedicated to research on flexible rotor systems with anisotropic rotor material properties. In addition, the anisotropy of rotor supports alters the rotor system resonance frequencies and the machine has to pass till it attains the operating angular speed. This phenomenon of rotor vibration is observed in vertical rotors. The aim of this work is to compare experimental vibration measurements and results of theoretical modeling. In the paper theoretical model, created from physical one of really existing rotor system is described. Collected experimental data of rotor vibrations in bearings are compared with results of theoretically derived equations. The results of theoretical modeling and research enables for estimation of a more precise technical condition of the rotor system both after the overhaul and during the maintenance and thus to avoid unexpected breakdowns, especially concerning the fatigue development in ball bearing elements.

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Khang, Nguyen Van, und Tran Van Luong. „On a programme for the balancing calculation of flexible rotors with the influence coefficient method“. Vietnam Journal of Mechanics 22, Nr. 4 (30.12.2000): 235–47. http://dx.doi.org/10.15625/0866-7136/9980.

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This paper presents the influence coefficient method of determining the locations of unbalances on a flexible rotor system and the correction weights. A computer software for calculating the at-the-site balancing of a flexible rotor system was created using C++ language at the Hanoi University of Technology. This software can be used by balancing flexible rotors in Vietnam.

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3

Maslen, E. H., und P. E. Allaire. „Magnetic Bearing Sizing for Flexible Rotors“. Journal of Tribology 114, Nr. 2 (01.04.1992): 223–29. http://dx.doi.org/10.1115/1.2920877.

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Magnetic bearing load capacity in flexible rotor systems must be adequate to accommodate external loads acting on the rotor. The first part of this paper develops the theoretical basis for computing the bearing load capacity requirements of flexible rotors subject to bounded external harmonic loads and strict motion constraints. The second part of this work illustrates the application of the theory to a specific example: a fairly simple three disk flexible rotor subject to mass unbalance loads. Load capacity requirements are computed for the example shaft at its first three free-free forward whirl critical speeds. Based on the bounds obtained, two bearing design cases are examined: one with 45 N load capacity and the other with 223 N load capacity. The performance of the rotor is not improved with the higher capacity bearing and neither is capable of adequately constraining the rotor at its second critical speed. It is concluded that this shaft cannot be operated above its second free-free critical speed without a midspan bearing.

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4

El-Shafei, A., A. S. El-Kabbany und A. A. Younan. „Rotor Balancing Without Trial Weights“. Journal of Engineering for Gas Turbines and Power 126, Nr. 3 (01.07.2004): 604–9. http://dx.doi.org/10.1115/1.1762903.

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The traditional balancing methods using trial or calibration weights are quite effective, yet too many trials may result in a lengthy balancing process. It had been suggested in the literature that it is possible to balance flexible rotors without the use of trial weights, if a rotor model is available. A procedure is developed in this paper to balance flexible rotors using complex modes and complex vibration measurements. It is shown that a complex rotor model is essential for the success of the technique. Moreover, careful calibration of the rotor model is the major cornerstone of the procedure. Experimental results illustrate the success of the procedure.

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Ma, Xunxun, Shujia Li, Wangliang Tian, Xiqiang Qu, Shengze Wang und Yongxing Wang. „Dynamic Behavior Analysis of the Winding Rotor with Structural Coupling and Time-Frequency Varying Parameters: Simulation and Measurement“. Applied Sciences 11, Nr. 17 (01.09.2021): 8124. http://dx.doi.org/10.3390/app11178124.

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To satisfy the requirements of high speed, large capacity and constant winding, a textile winding rotor needs to work in a wide rotation speed range and frequently pass through critical speed points. Thus, the winding rotor adopts the flexible long shaft coupling structure and flexible support with rubber O-rings. This kind of rotor has a multi-coupling structure and frequency-dependent parameters characteristics, especially representative and universal in the dynamic analysis method of the high-speed rotor. In this paper, an approach was proposed to investigate the dynamic behavior of the winding rotor considering the flexible coupling and frequency-dependent supporting parameters. Firstly, a dynamic model of the winding rotor was established by using a Timoshenko beam element. Its dynamic behaviors were simulated by considering the time-varying rotation speed and the frequency-dependent parameters of flexible support. Secondly, a non-contact measuring device was developed for measuring the vibration displacement of the winding rotor in three different speed-up times. Finally, based on simulation and measurement data, how flexible support parameters and the speed-up time affect the winding rotor passing through the critical speed point of the rotor smoothly is revealed. The methods and findings reported here can be used for theoretical and experimental vibration analysis of other types of high-speed flexible rotors.

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Muhammed Ameen, Yahya, und Jaafar Khalaf Ali. „Flexible rotor balancing without trial runs using experimentally tuned FE based rotor model“. Basrah journal of engineering science 21, Nr. 1 (01.02.2021): 20–26. http://dx.doi.org/10.33971/bjes.21.1.4.

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A method based on experimentally calibrated rotor model is proposed in this work for unbalance identification of flexible rotors without trial runs. Influence coefficient balancing method especially when applied to flexible rotors is disadvantaged by its low efficiency and lengthy procedure, whilst the proposed method has the advantage of being efficient, applicable to multi-operating spin speeds and do not need trial runs. An accurate model for the rotor and its supports based on rotordynamics and finite elements analysis combined with experimental modal analysis, is produced to identify the unbalance distribution on the rotor. To create digital model of the rotor, frequency response functions (FRFs) are determined from excitation and response data, and then modal parameters (natural frequencies and mode shapes) are extracted and compared with experimental analogies. Unbalance response is measured traditionally on rotor supports, in this work the response measured from rotating disks instead. The obtained results show that the proposed approach provides an effective alternative in rotor balancing. Increasing the number of balancing disks on balancing quality is investigated as well.

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Zaytsev, Nikolay, Denis Zaytsev, Andrey Makarov und Dmitriy Mineev. „NUMERICAL SIMULATION OF THE DYNAMICS OF A FLEXIBLE ROTOR WITH TWO BALL AUTO-BALANCERS“. Perm National Research Polytechnic University Aerospace Engineering Bulletin, Nr. 62 (2020): 31–44. http://dx.doi.org/10.15593/2224-9982/2020.62.04.

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Ball auto-balancing devices can to compensate changes of unbalance "on the move" only for rotors operating at supercritical speeds. For automatic balancing of such rotors, classified as flexible rotors, several auto-balancers located in different cross sections of the shaft are necessary. This makes it necessary to account bending fluctuations on studies of dynamics of the rotor with auto-balancers, that is especially important in the design of the real rotors. In view of the complexity of experimental studies of such rotors in the article the method of direct numerical simulation of the dynamics of the flexible rotor system – supports – auto-balances is considered. The methodological basis of this method is the use of a discrete multi-mass rotor model, which is equivalent in dynamic characteristics to a real rotor, and also the equations of dynamics of the system discrete rotor – supports – auto-balancers, obtained in the direct form of recording. For definition of discrete masses and a matrix of coefficients of influence of stiffness of rotor cross-sections it is supposed to use calculations for finite-element model of a real rotor by existing software complexes of the engineering analysis. The mathematical model of the system dynamics obtained by the Lagrange method takes into account the non-stationarity of the rotor rotation speed, the influence of gravity and the rolling friction of the balls in the auto-balancer cages. Verification of the mathematical model was performed by reproducing the published data using a computational model for a two-support single-disk three-mass rotor with a two-ball auto-balancer. For a four-mass rotor with two two-ball auto-balancers, the results of numerical simulation of dynamics for the modes of acceleration, steady-state rotation and deceleration are presented. It is shown that for the system under consideration, only partial auto-balancing takes place in the steady rotation mode, including after a stepwise increase of the imbalance.

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Halder, B., A. Mukherjee und R. Karmakar. „Theoretical and Experimental Studies on Squeeze Film Stabilizers for Flexible Rotor-Bearing Systems Using Newtonian and Viscoelastic Lubricants“. Journal of Vibration and Acoustics 112, Nr. 4 (01.10.1990): 473–82. http://dx.doi.org/10.1115/1.2930131.

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A combination of a squeeze film damper and a plane journal bearing is studied as a stabilizing scheme. The damper is made to play the role of a stabilizer to postpone the instability threshold speeds of flexible rotors. Both Newtonian and viscoelastic fluids are used in the rotor-bearing system. Dynamics of the system is theoretically analyzed using bond graphs. Analysis reveals that the use of a Newtonian fluid in the stabilizer largely improves the high speed stability range. However, viscoelastic stabilizing fluid has a detrimental effect on highly flexible rotors. Experimental investigations, conducted on a flexible rotor (natural frequency, 30 Hz), confirm the theoretical findings. In addition, experiments indicate that though the use of viscoelastic stabilizing fluids leads to instability in flexible rotors, the growth of large amplitude whirl is postponed to very high speeds.

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Smalley, Anthony J. „Jørgen Lund: A Perspective on His Contributions to Modern Rotor Bearing Dynamics“. Journal of Vibration and Acoustics 125, Nr. 4 (01.10.2003): 434–40. http://dx.doi.org/10.1115/1.1605765.

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This paper presents a perspective on the work of Dr. J. W. Lund, in the field of rotor-bearing dynamics. It traces his most influential work in published journals, software, data published in various manuals, government reports, and course notes. It addresses his work in oil bearings, gas bearings, unbalance response of flexible rotors, stability of flexible rotors, squeeze-film dampers, rotor balancing, and modal testing of rotors. It provides a road map for those who would revisit the knowledge base. It shows how Dr. Lund’s work has permeated the practice of today’s suppliers and users of turbomachinery, and industry consultants.

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Keogh, P. S., und M. O. T. Cole. „Rotor vibration with auxiliary bearing contact in magnetic bearing systems Part 1: Synchronous dynamics“. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 217, Nr. 4 (01.04.2003): 377–92. http://dx.doi.org/10.1243/095440603321509676.

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Magnetic bearing systems incorporate auxiliary bearings to prevent physical interaction between rotor and stator laminations. Rotor/auxiliary bearing contacts may occur when a magnetic bearing still retains a full control capability. To actively return the rotor to a non-contacting state it is essential to determine the manner in which contact events affect the rotor vibration signals used for position control. An analytical procedure is used to assess the nature of rotor contact modes under idealized contacts. Non-linearities arising from contact and magnetic bearing forces are then included in simulation studies involving rigid and flexible rotors to predict rotor response and evaluate rotor synchronous vibration components. An experimental flexible rotor/magnetic bearing facility is also used to validate the predictions. It is shown that changes in synchronous vibration amplitude and phase induced by contact events causes existing controllers to be ineffective in attenuating rotor displacements. These findings are used in Part 2 of the paper as a foundation for the design of new controllers that are able to recover rotor position control under a range of contact cases.

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Urbiola-Soto, Leonardo, und Marcelo Lopez-Parra. „Liquid Self-Balancing Device Effects on Flexible Rotor Stability“. Shock and Vibration 20, Nr. 1 (2013): 109–21. http://dx.doi.org/10.1155/2013/742163.

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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.

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12

Schlotter, Michael, und Patrick S. Keogh. „Synchronous Position Recovery Control for Flexible Rotors in Contact with Auxiliary Bearings“. Journal of Vibration and Acoustics 129, Nr. 5 (02.02.2007): 550–58. http://dx.doi.org/10.1115/1.2731414.

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This paper details a methology for the active recovery of contact free levitation of a rotor from a state of persistent contact with auxiliary bearings. An analytical method to describe contact dynamics of flexible rotors is presented. It shows that synchronous unbalance forces can cause a rotor to adopt stable contact modes, which are characterized by periodic motion and a fixed contact point in a rotating frame of reference. Based on these observations, a recovery strategy is developed to return the rotor to a contact free state. Compensation forces may be applied by magnetic bearings to reduce the effective synchronous forcing which is driving the contact, so that the rotor can progress to a contact free orbit. It is shown that even in the presence of highly nonlinear contact dynamic effects, a linear finite element rotor model can be used to calculate appropriate influence coefficients. The contact recovery procedure is successfully verified by simulations and measurements on a flexible rotor test facility. Allowable bounds on the phase of the synchronous recovery forces are investigated and limitations of the method are discussed.

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Shen, I. Y., und Hyunchul Kim. „A Linearized Theory on Ground-Based Vibration Response of Rotating Asymmetric Flexible Structures“. Journal of Vibration and Acoustics 128, Nr. 3 (09.01.2006): 375–84. http://dx.doi.org/10.1115/1.2172265.

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This paper is to develop a unified algorithm to predict vibration of spinning asymmetric rotors with arbitrary geometry and complexity. Specifically, the algorithm is to predict vibration response of spinning rotors from a ground-based observer. As a first approximation, the effects of housings and bearings are not included in this analysis. The unified algorithm consists of three steps. The first step is to conduct a finite element analysis on the corresponding stationary rotor to extract natural frequencies and mode shapes. The second step is to represent the vibration of the spinning rotor in terms of the mode shapes and their modal response in a coordinate system that is rotating with the spinning rotor. The equation of motion governing the modal response is derived through use of the Lagrange equation. To construct the equation of motion, explicitly, the results from the finite element analysis will be used to calculate the gyroscopic matrix, centrifugal stiffening (or softening) matrix, and generalized modal excitation vector. The third step is to solve the equation of motion to obtain the modal response, which, in turn, will lead to physical response of the rotor for a rotor-based observer or for a ground-based observer through a coordinate transformation. Results of the algorithm indicate that Campbell diagrams of spinning asymmetric rotors will not only have traditional forward and backward primary resonances as in axisymmetric rotors, but also have secondary resonances caused by higher harmonics resulting from the mode shapes. Finally, the algorithm is validated through a calibrated experiment using rotating disks with evenly spaced radial slots. Qualitatively, all measured vibration spectra show significant forward and backward primary resonances as well as secondary resonances as predicted in the theoretical analysis. Quantitatively, measured primary and secondary resonance frequencies agree extremely well with those predicted from the algorithm with mostly <3.5% difference.

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14

Hui, Han, Li Na Hao, Zhang Qi und Gao Xiang. „The Trapezoid Counterweight Method in Dynamic Balance of Symmetric Flexible Rotor“. Applied Mechanics and Materials 868 (Juli 2017): 201–6. http://dx.doi.org/10.4028/www.scientific.net/amm.868.201.

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Steam turbine generator unit, water pump and other high speed revolution symmetric flexible rotor were regarded as research objects in this paper. According to variation characteristic of rotor shaft in rigid and flexible working mode, nine-reel high pressure water pump rotor was analyzed. The former four-order intrinsic frequency of flexible rotor was obtained by modal analysis and harmonic response analysis. The methods of reaction force response and unbalance response were been studied after first order and second order resonance region eliminating in different modes of simple harmonic exciting force. Based on above theoretical research results, trapezoid counterweight method was proposed for dynamic balance of flexible rotor. This method solved problem that rigid dynamic balance of low speed rotor was destroyed after first order and second order resonance region counter weight in dynamic balance of flexible rotor. The dynamic balancing techniques of flexible rotor could be improved the qualities of rotor and its relative products by this method, eliminating the vibration of unbalance mass of products radically.

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Khulief, Y. A., M. A. Mohiuddin und M. El-Gebeily. „A New Method for Field-Balancing of High-Speed Flexible Rotors without Trial Weights“. International Journal of Rotating Machinery 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/603241.

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Flexible rotor balancing, in general, relies to a great extent on physical insight into the modal nature of the unbalance response. The objective of this investigation is to develop a hybrid experimental/analytical technique for balancing high-speed flexible rotors. The developed technique adopts an approach that combines the finite element modeling, experimental modal analysis, vibration measurements, and mathematical identification. The modal imbalances are identified and then transformed to the nodal space, in order to determine a set of physical balancing masses at some selected correction planes. The developed method does not rely on trial runs. In addition, the method does not require operating the supercritical rotor in a high-speed balancing facility, while accounting for the contribution of higher significant modes. The developed scheme is applied to a multidisk, multibearing, high-speed flexible rotor, where the interaction between the rotor-bending operating deflections and the forces resulting from the residual unbalance are appreciable. Some new benchmark solutions and observations are reported. The applicability, reliability, and challenges that may be encountered in field applications are addressed.

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Va´zquez, Jose´ A., Lloyd E. Barrett und Ronald D. Flack. „A Flexible Rotor on Flexible Bearing Supports: Stability and Unbalance Response“. Journal of Vibration and Acoustics 123, Nr. 2 (01.12.2000): 137–44. http://dx.doi.org/10.1115/1.1355244.

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An experimental study of the effects of bearing support flexibility on rotor stability and unbalance response is presented. A flexible rotor supported by fluid film bearings on flexible supports was used with fifteen support configurations. The horizontal support stiffness was varied systematically while the vertical stiffness was kept constant. The support characteristics were determined experimentally by measuring the frequency response functions of the support structure at the bearing locations. These frequency response functions were used to calculate polynomial transfer functions that represented the support structure. Stability predictions were compared with measured stability thresholds. The predicted stability thresholds agree with the experimental data within a confidence bound for the logarithmic decrement of ±0.01. For unbalance response, the second critical speed of the rotor varied from 3690 rpm to 5200 rpm, depending on the support configuration. The predicted first critical speeds agree with the experimental data within −1.7 percent. The predicted second critical speeds agree with the experimental data within 3.4 percent. Predictions for the rotor on rigid supports are included for comparison.

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17

Park, J. S., und S. N. Jung. „Comprehensive multibody dynamics analysis for rotor aeromechanics predictions in descending flight“. Aeronautical Journal 116, Nr. 1177 (März 2012): 229–49. http://dx.doi.org/10.1017/s0001924000006813.

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AbstractThis paper studies the rotor aeromechanics in descending flight using a nonlinear flexible multibody dynamic analysis code, DYMORE. A freewake model is included in DYMORE to improve the rotor wake modelling. The wind-tunnel test data of the Higher-harmonic Aeroacoustics Rotor Test (HART) II rotor, with and without higher harmonic pitch control (HHC), and the flight test data of the full-scale utility helicopter rotor in descent are used for the aeromechanics correlation at an advance ratio of 0·15. The blade-vortex interaction (BVI) airloads are reasonably predicted for both the HART II and utility helicopter rotors, although some BVI peaks are missed on the advancing sides for both the rotors. The flap deflections and elastic torsion deformations at the blade tip are fairly correlated against the measured data of the HART II rotor. The correlation of blade structural moments for both HART II and utility helicopter rotors are not as good as the lift predictions; however, a reasonable prediction is obtained for the utility helicopter rotor.

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Mukherjee, Abhro, Satyabrata Das und Sabyasachi S. Sengupta. „Whirl Orbital Response Control of Micro Rotors With Flexural Modes“. International Journal of Acoustics and Vibration 23, No 3, September 2018 (September 2018): 343–54. http://dx.doi.org/10.20855/ijav.2018.23.31180.

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In this paper there is an attempt to develop an elegant controlling strategy to stabilize flexible rotors having flexural modes. Initially, the possible causes of instability in a rotor system due to internal damping through mathematical formulation are discussed. The threshold or the critical speed was derived beyond which the rotor became unstable. The stabilizing algorithm was designed by keeping the overall control law mathematically simple such that minimum numbers of sensors were used and deployed. Actuators were properly selected such that control and stabilization was practically possible for rotors of such low diameters. Bond graphs were used to model the rotor system having flexural modes. As bond graphs act as a portrayal of power and information exchange within the system and with external environment, bond graphs may give a better clue for modeling such systems. The real challenge in this work lies in the proposal of control strategy in stabilizing non-rigid rotors having flexible modes. Mathematical simplicity of the proposed control algorithms is considered for its easy deployment. The Simplest Beam model the Euler-Bernoulli beam is considered.

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Sinou, J. J., C. Villa und F. Thouverez. „Experimental and Numerical Investigations of a Flexible Rotor on Flexible Bearing Supports“. International Journal of Rotating Machinery 2005, Nr. 3 (2005): 179–89. http://dx.doi.org/10.1155/ijrm.2005.179.

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The main goal of this paper is to study, numerically and experimentally, the effects of the bearing support flexibility on the rotor dynamic and the first forward and backward critical speeds. The test rig which is used in this study has been developed and built at the École Centrale de Lyon (France). This flexible rotor, supported by two rolling bearings on flexible supports, is used for three configurations of the flexible supports. The support characteristics are determined experimentally by performing static tests. Moreover, a finite element model of this flexible rotor is presented which consists of a rigid disk on a flexible shaft supported by two bearing supports. On the basis of measured frequency response functions for various rotational speeds, eigenfrequencies and the associated Campbell diagram from the numerical model and the related experimental results for the flexible rotor are discussed. The comparison of these experimental and numerical tests are used in order to update the finite element model and the associated moment stiffness of the two rolling bearings for the three configurations of interest.

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Qiao, Xiaoli, und Guojun Hu. „Active Control for Multinode Unbalanced Vibration of Flexible Spindle Rotor System with Active Magnetic Bearing“. Shock and Vibration 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/9706493.

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The unbalanced vibration of the spindle rotor system in high-speed cutting processes not only seriously affects the surface quality of the machined products, but also greatly reduces the service life of the electric spindle. However, since the unbalanced vibration is often distributed on different node positions, the multinode unbalanced vibration greatly exacerbates the difficulty of vibration control. Based on the traditional influence coefficient method for controlling the vibration of a flexible rotor, the optimal influence coefficient control method with weights for multinode unbalanced vibration of flexible electric spindle rotors is proposed. The unbalanced vibration of all nodes on the whole spindle rotor is used as the control objective function to achieve optimal control. The simulation results show that the method has an obvious control effect on multinode unbalanced vibration.

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Lee, A.-C., und Y.-P. Shih. „Identification of the Unbalance Distribution and Dynamic Characteristics of Bearings in Flexible Rotors“. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 210, Nr. 5 (September 1996): 409–32. http://dx.doi.org/10.1243/pime_proc_1996_210_216_02.

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This paper presents a new method for estimating dynamic characteristics of bearings and unbalance distribution in flexible rotors based on the transfer matrix method for analysing the steady state responses of rotor-bearing systems, in which rotary inertia, gyroscopic and transverse-shear effects are also considered. Identification can be realized from the measured response data influenced by bearing characteristics and rotor unbalance using the least-squares method. Justification of the method is presented by numerical simulations.

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Williams, R. D., F. J. Keith und P. E. Allaire. „A Comparison of Analog and Digital Controls for Rotor Dynamic Vibration Reduction Through Active Magnetic Bearings“. Journal of Engineering for Gas Turbines and Power 113, Nr. 4 (01.10.1991): 535–43. http://dx.doi.org/10.1115/1.2906274.

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Active magnetic bearings are implemented using analog and digital controllers to achieve vibration reduction for multimass flexible rotors. Various models are developed for the rotor-bearing system, and the first three critical speeds (resonant frequencies) are shown to be unaffected by inclusion of the higher order shaft dynamics in the model. Higher order rotor-bearing models reveal the presence of “shaft modes,” the excitation of which is a function of the position of the magnetic bearing proximity probe (these modes are effectively damped by the flexible coupling employed in the test apparatus). Rotor dynamic behavior is investigated for various analog and digital controllers. Rotor response in the presence of proportional-derivative control is similar for both analog and digital control. Higher order digital control algorithms (second derivative and integral) affect the rotor response in a frequency dependent manner: Second derivative feedback is effective at reducing third mode vibration, and integral feedback, while rejecting any steady-state rotor position error, slightly accentuates the vibration at the first critical speed. Increasing the sampling rate of the digital controller has a similar effect to increasing the amount of second derivative feedback employed.

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Tonnesen, Jørgen. „Jørgen W. Lund: Theories Versus Tests, Part 1: Balancing and Response of Flexible Rotors“. Journal of Vibration and Acoustics 125, Nr. 4 (01.10.2003): 482–88. http://dx.doi.org/10.1115/1.1606693.

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The contribution of the late Professor Jørgen W. Lund in the field of rotor dynamics and fluid film bearings is presented in a condensed form with the emphasis on the experimental work and results that confirm and support many of Dr. Lund’s theories and analyses. Part 1(A) covers the subject of rotor balancing by the influence method and Part 1(B) the unbalance response of a flexible rotor. In Part 2(A), experiments on instability and the influence of unbalance on whirl frequency are discussed for two rotors supported in cylindrical bearings. In Part 2(B), experiments using a cylindrical, two-axial groove journal bearing over a range of loads and speeds are reviewed.

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Ran, Shaolin, Yefa Hu, Huachun Wu und Xin Cheng. „Resonance Vibration Control for AMB Flexible Rotor System Based onµ-Synthesis Controller“. Mathematical Problems in Engineering 2018 (19.12.2018): 1–16. http://dx.doi.org/10.1155/2018/4362101.

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The resonance vibration control of flexible rotor supported on active magnetic bearings (AMB) is a challenging issue in the industrial applications. This work addresses the application of robust control method to the resonance vibration control for AMB flexible rotor while passing through the critical speed. This model-based method shows great superiority to handling flexible mode vibration, which can guarantee robust stability and performance when encountering modal perturbation. First, the designed flexible rotor-AMB test rig is briefly introduced. Then the system modeling is described in detail including flexible rotor, power amplifier, displacement sensors and magnetic actuator and rotordynamics are analyzed. Model validation is carried out by sine sweeping test. Finally, theμ-synthesis controller is designed. The simulation and experimental results indicate that the designedμ-synthesis controller, which shows great robustness to modal perturbation, can effectively suppress the resonance vibration of flexible rotor and achieve supercritical operation.

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Maslen, E., und D. Montie. „Sliding Mode Control of Magnetic Bearings: A Hardware Perspective“. Journal of Engineering for Gas Turbines and Power 123, Nr. 4 (01.03.1999): 878–85. http://dx.doi.org/10.1115/1.1383771.

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The physical hardware of magnetic bearing systems has a very strong impact on the application of sliding mode control to this technology. Finite current slew rates affect the character of achievable reaching conditions; flexible rotors demand the use of state observers which then affect achievable robustness; finite controller throughput and use of switching power amplifiers has substantial impact on how chatter should be addressed. This paper formulates the sliding mode control problem for realistic rotor/magnetic actuator systems considering rotor flexibility and finite amplifier voltages. The structure of the resulting class of controllers is examined for both rigid and for flexible rotors. It is demonstrated that, fundamentally, these controllers are conventional linear state observer-based controllers acting through high-speed switching amplifiers as in conventional magnetic bearing technology.

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Sinou, Jean-Jacques, und Fabrice Thouverez. „Experimental Study of a Flexible Rotor and Its Dependency on the Rolling-Bearing Temperature“. International Journal of Rotating Machinery 2006 (2006): 1–8. http://dx.doi.org/10.1155/ijrm/2006/38595.

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An experimental study of the effects of the bearing temperature on rotor unbalance responses and on the first forward and backward critical speeds is presented. Experimental tests consisted in run-ups and run-downs are realized to a flexible rotor developed and built at the Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS 5513, École Centrale de Lyon, France. This flexible rotor is supported by two rolling bearings on flexible supports. It will be demonstrated that rolling-element bearing temperature has no-negligible effects of the amplitude and values of the forward and backward first critical speeds of the flexible rotor. In addition to the investigation of the influences of the bearing temperature on the dynamics of the flexible rotor at the first critical speeds, a brief investigation into the complete dynamic within the speed range of interest is also conducted. The purpose of this is to examine in detail the vibrational phenomena of the flexible rotor and to detect some characteristics dynamic behaviour due to the presence of rolling-element bearings.

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Ran, Shaolin, Yefa Hu, Huachun Wu und Xin Cheng. „Active vibration control of the flexible high-speed rotor with magnetic bearings via phase compensation to pass critical speed“. Journal of Low Frequency Noise, Vibration and Active Control 38, Nr. 2 (18.12.2018): 633–46. http://dx.doi.org/10.1177/1461348418819404.

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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.

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Deng, Hao, Xi Fang, Huachun Wu, Yiming Ding, Jinghu Yu, Xiaoyu Zhang, Hong Li und Chenwei Liu. „Dynamic Analysis of Flexible Rotor Based on Transfer Symplectic Matrix“. Shock and Vibration 2019 (09.09.2019): 1–11. http://dx.doi.org/10.1155/2019/9154272.

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In view of the numerical instability and low accuracy of the traditional transfer matrix method in solving the high-order critical speed of the rotor system, a new idea of incorporating the finite element method into the transfer matrix is proposed. Based on the variational principle, the transfer symplectic matrix of gyro rotors suitable for all kinds of boundary conditions and supporting conditions under the Hamilton system is derived by introducing dual variables. To verify the proposed method in rotor critical speed, a numerical analysis is adopted. The simulation experiment results show that, in the calculation of high-order critical speed, especially when exceeding the sixth critical speed, the numerical accuracy of the transfer symplectic matrix method is obviously better than that of the reference method. The relative errors between the numerical solution and the exact solution are 0.0347% and 0.2228%, respectively, at the sixth critical speed. The numerical example indicates the feasibility and superiority of the method, which provides the basis for the optimal design of the rotor system.

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Chen, H. Ming. „Magnetic Bearings and Flexible Rotor Dynamics“. Tribology Transactions 32, Nr. 1 (Januar 1989): 9–15. http://dx.doi.org/10.1080/10402008908981856.

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30

Krodkiewski, J. M., Y. Cen und L. Sun. „Improvement of Stability of Rotor System by Introducing a Hydraulic Damper into an Active Journal Bearing“. International Journal of Rotating Machinery 3, Nr. 1 (1997): 45–52. http://dx.doi.org/10.1155/s1023621x97000055.

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Modelling and analysis of a rotor--bearing system with a new type of active oil bearing is presented. The active bearing is supplied with a flexible sleeve whose deformation can be changed during operation of the rotor. The flexible sleeve is also a part of a hydraulic damper whose parameters can be controlled during operation as well.Finite Element Method (FEM) and the Guyan condensation technique was utilised to create mathematical model of both, the rotor and the flexible sleeve. The hydrodynamic pressure distribution in the oil film, for the instantaneous position of the flexible sleeve and rotor, was approximated by Reynolds equation.The mathematical model of motion of a rotor system with the described active bearing developed in this paper allowed the influence of the introduced hydraulic damper on stability of the rotor-bearing system to be investigated. Results of the computer simulation shows that within a large region of configuration parameters of the rotor bearing system, the self exciting vibration can be eliminated or greatly reduced during operation by properly controlled deformation of the flexible sleeve and optimal choice of the hydraulic damper parameters.

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Ishimatsu, Takakazu, Takashi Shimomachi und Nobuyoshi Taguchi. „Active Control for Whirling Motion of Flexible Rotor“. Journal of Robotics and Mechatronics 3, Nr. 4 (20.08.1991): 360–64. http://dx.doi.org/10.20965/jrm.1991.p0360.

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In a rotational machine, unbalance on the rotor is formidable since it causes resonance synchronized with the rotation of the rotor. In order to suppress this unfavorable vibration, we built an active vibration control system of flexible rotor using an electromagnetic damper. Our control system is composed of a digital controller to suppress the rotationally synchronized whirling motion and also a conventional analogue controller. Using our control system, whirling motion of the rotor under various rotating speeds was suppressed significantly.

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Wileman, J., und I. Green. „Stability Analysis of Mechanical Seals With Two Flexibly Mounted Rotors“. Journal of Tribology 120, Nr. 2 (01.04.1998): 145–51. http://dx.doi.org/10.1115/1.2834401.

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Dynamic stability is investigated for a mechanical seal configuration in which both seal elements are flexibly mounted to independently rotating shafts. The analysis is applicable to systems with both counterrotating and corotating shafts. The fluid film effects are modeled using rotor dynamic coefficients, and the equations of motion are presented including the dynamic properties of the flexible support. A closed-form solution for the stability criteria is presented for the simplifled case in which the support damping is neglected. A method is presented for obtaining the stability threshold of the general case, including support damping. This method allows instant determination of the stability threshold for a fully-defined seal design. A parametric study of an example seal is presented to illustrate the method and to examine the effects of various parameters in the seal design upon the stability threshold. The fluid film properties in the example seal are shown to affect stability much more than the support properties. Rotors having the form of short disks are shown to benefit from gyroscopic effects which give them a larger range of stable operating speeds than long rotors. For seals with one long rotor, counterrotating operation is shown to be superior because the increased fluid stiffness transfers restoring moments from the short rotor to the long.

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Bortolotti, Pietro, Nick Johnson, Nikhar J. Abbas, Evan Anderson, Ernesto Camarena und Joshua Paquette. „Land-based wind turbines with flexible rail-transportable blades – Part 1: Conceptual design and aeroservoelastic performance“. Wind Energy Science 6, Nr. 5 (23.09.2021): 1277–90. http://dx.doi.org/10.5194/wes-6-1277-2021.

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Abstract. This work investigates the conceptual design and the aeroservoelastic performance of land-based wind turbines whose blades can be transported on rail via controlled bending. The turbines have a nameplate power of 5 MW and a rotor diameter of 206 m, and they aim to represent the next generation of land-based machines. Three upwind designs and two downwind designs are presented, combining different design goals together with conventional glass and pultruded carbon fiber laminates in the spar caps. One of the five blade designs is segmented and serves as a benchmark to the state of the art in industry. The results show that controlled flexing requires a reduction in the flapwise stiffness of the blades, but it represents a promising pathway for increasing the size of land-based wind turbine rotors. Given the required stiffness, the rotor can be designed either downwind with standard rotor preconing and nacelle uptilt angles or upwind with higher-than-usual angles. A downwind-specific controller is also presented, featuring a cut-out wind speed reduced to 19 m s−1 and a pitch-to-stall shutdown strategy to minimize blade tip deflections toward the tower. The flexible upwind and downwind rotor designs equipped with pultruded carbon fiber spar caps are found to generate the lowest levelized cost of energy, 2.9 % and 1.3 %, respectively, less than the segmented design. The paper concludes with several recommendations for future work in the area of large flexible wind turbine rotors.

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Lepore, Francisco Paulo, Marcelo Braga Santos und Rafael Gonçalves Barreto. „Identification of Rotary Machines Excitation Forces Using Wavelet Transform and Neural Networks“. Shock and Vibration 9, Nr. 4-5 (2002): 235–44. http://dx.doi.org/10.1155/2002/967638.

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Unbalance and asynchronous forces acting on a flexible rotor are characterized by their positions, amplitudes, frequencies and phases, using its measured vibration responses. The rotary machine dynamic model is a neural network trained with measured vibration signals previously decomposed by wavelets. A typical compaction ratio of 2048:4 is achieved in this application, considering the stationary nature of the measured vibrations signals and the shape of the chosen wavelet function. The Matching Pursuit procedure, coupled to a modified Simulated Annealing optimization algorithm is used to decompose the vibration signals. The performance of several neural network with different input database sets is analyzed to define the best network architecture in the sense to achieve successful training, minimum identification error, with maximum probability to give the correct answers. The experiments are conducted on a vertical rotor with three rigid discs mounted on a flexible shaft supported by two flexible bearings. The vibration responses are measured at the bearings and at the discs. A methodology to balance flexible rotors based on the proposed identification methodology is also presented.

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Li, Liqing, Yuanhang Hou und Shuqian Cao. „An Optimized Modal Balancing Approach for a Flexible Rotor Using a Vibration Response While the Rotor Is Speeding Up“. Shock and Vibration 2022 (07.03.2022): 1–13. http://dx.doi.org/10.1155/2022/5261279.

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The modal balancing method (MBM) is an effective method for reducing the vibration caused by the unbalancing of a rotor. A rotor’s modal parameters, especially its modal shapes, need to be accurately calculated in this method. This paper proposes an optimized modal balancing approach for flexible rotors. The vibration modes of the rotor are first obtained with experimental modal analysis based on the rotor’s response signals while the rotor is speeding up. The rotor balancing strategy is subsequently optimized by the sensitivity analysis of the mode shapes. Orthogonal trial masses are obtained based on the orthogonality of each vibration mode, and the correction masses are finally calculated by using the influence coefficients of the trial masses. An experimental result is shown to demonstrate and validate that the proposed approach is able to achieve superior accuracy compared to the conventional MBM.

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Choy, F. K., J. Padovan und C. Batur. „Rub Interactions of Flexible Casing Rotor Systems“. Journal of Engineering for Gas Turbines and Power 111, Nr. 4 (01.10.1989): 652–58. http://dx.doi.org/10.1115/1.3240308.

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Rub interactions between a rotor assembly and its corresponding casing structure has long been one of the major causes for machine failure. Fracture/fatigue failures of turbine impeller blade components may even lead to catastrophic consequences. This paper presents a comprehensive analysis of a complex rotor-bearing-blade-casing system during component rub interactions. The modal method is used in this study. Orthonormal coupled rotor-casing modes are used to obtain accurate relative motion between rotor and casing. External base vibration input and the sudden increase of imbalance are used to simulate suddenly imposed adversed operating condition. Nonlinear turbine/impeller blade effects are included with the various stages of single/multiple blade participation. A variable integration time step procedure is introduced to insure both accuracy and efficiency in numerical solutions. The dynamic characteristics of the system are examined in both the time domain and the frequency domain using a numerical FFT procedure. Nonlinear bearing and seal forces are also included to enhance a better simulation of the operating system. Frequency components of the system spectral characteristics will be correlated with the localized rub excitations to enable rub signature analysis. A multibearing flexible casing rotor system will be used as an example. Conclusions will be drawn from the results of an extensive parametric study.

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Li, Zi Gang, und Ming Li. „Non-Linear Dynamics of a Flexible Multi-Rotor Bearing System with a Fault of Parallel Misalignment“. Applied Mechanics and Materials 138-139 (November 2011): 104–10. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.104.

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The dynamic behaviors of a flexible multi-rotor system with a fault of parallel misalignment are investigated on the basis of assumptions, such as the long journal bearings, small rotor misalignment and mass disk unbalance. Firstly, based on the Lagrange equations with undetermined multiplier, the dynamic model of a rotor system under the action of the nonlinear oil film forces is developed after taking into account the holonomic constraint, which describes the misalignment relation between two rotors, and the theoretical analysis reveals that the system with eleven DOF is of strong nonlinear properties. Then the nonlinear dynamic characteristics on numerical technique, such as steady state response, rotor orbit, Poincaré section and the largest Lyapunov exponent, are paid more attention in this study. The results show that at low speed the components of the steady-state responses in lateral direction is of the synchronous frequency with rotating speed as well as its integer multiples frequencies. As the speed increases the dynamic characteristics become complicated, and the nT-period, quasi-period and chaotic oscillations occur.

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38

Wang, Meiling, Qingkai Han, Baogang Wen, Hao Zhang und Tianmin Guan. „Modal characteristics and unbalance responses of fan rotor system with flexible support structures in aero-engine“. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, Nr. 9 (12.07.2016): 1686–705. http://dx.doi.org/10.1177/0954410016658076.

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This paper investigates the vibration patterns, i.e. rigid motions of shaft and elastic deformation of support structures, of fan rotor system in aero-engine, which differs from traditional flexible rotor systems, and together with its shaft transverse motions due to unbalanced mass. The fan rotor system commonly is composed of one rigid shaft and two flexible support structures (such as squirrel cages), which is effective to decrease the critical speeds avoiding serious shaft vibration due to unbalance. Scaled test rig for realistic fan rotor system is set up according to similarity principles, governing differential equations of which are deduced by means of Lagrangian approach with four degrees of freedom. In contrast to modeling a traditional flexible rotor system, the system stiffness is not determined by the shaft but the two flexible support structures. The rigid shaft only contributes to the inertial items of the governing equations. Parameter values of dynamic model are identified from measurements on the scaled test rig, the modal shapes and the modal energy distributions are calculated. These modal characteristics of the fan rotor system are quite different from those of a traditional flexible rotor system whose stiffness mainly contributed by its elastic shaft even the system values are consistent. The obtained modal characteristics are compared and confirmed by using the simulation results of a corresponding finite element model, in which shaft is built by rotating beam elements and its flexible structures are built by equivalent spring elements. Campbell diagrams of the fan rotor system are used to illustrate the gyroscopic effect with the increasing speeds. And then the unbalance responses are calculated through the deduced analytical formula rapidly and comparisons, including the response spectrum and orbits, the amplitude and phase frequency response curves, and operating deflection shapes, are carried out in the sub- and super-critical range.

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Kurvinen, Emil, Raine Viitala, Tuhin Choudhury, Janne Heikkinen und Jussi Sopanen. „Simulation of Subcritical Vibrations of a Large Flexible Rotor with Varying Spherical Roller Bearing Clearance and Roundness Profiles“. Machines 8, Nr. 2 (29.05.2020): 28. http://dx.doi.org/10.3390/machines8020028.

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In large rotor-bearing systems, the rolling element bearings act as a considerable source of subcritical vibration excitation. Simulation of such rotor bearing systems contains major sources of uncertainty contributing to the excitation, namely the roundness profile of the bearing inner ring and the clearance of the bearing. In the present study, a simulation approach was prepared to investigate carefully the effect of varying roundness profile and clearance on the subcritical vibration excitation. The FEM-based rotor-bearing system simulation model included a detailed description of the bearings and asymmetricity of the rotor. The simulation results were compared to measured responses for validation. The results suggest that the simulation model was able to capture the response of the rotor within a reasonable accuracy compared to the measured responses. The bearing clearance was observed to have a major effect on the subcritical resonance response amplitudes. In addition, the simulation model confirmed that the resonances of the 3rd and 4th harmonic vibration components in addition to the well-known 2nd harmonic resonance (half-critical resonance) can be significantly high and should thus be taken into account already in the design phase of large subcritical rotors.

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Koroishi, Edson Hideki, Adriano Silva Borges, Aldemir Ap Cavalini und Valder Steffen. „Numerical and Experimental Modal Control of Flexible Rotor Using Electromagnetic Actuator“. Mathematical Problems in Engineering 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/361418.

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The present work is dedicated to active modal control applied to flexible rotors. The effectiveness of the corresponding techniques for controlling a flexible rotor is tested numerically and experimentally. Two different approaches are used to determine the appropriate controllers. The first uses the linear quadratic regulator and the second approach is the fuzzy modal control. This paper is focused on the electromagnetic actuator, which in this case is part of a hybrid bearing. Due to numerical reasons it was necessary to reduce the size of the model of the rotating system so that the design of the controllers and estimator could be performed. The role of the Kalman estimator in the present contribution is to estimate the modal states of the system and to determine the displacement of the rotor at the position of the hybrid bearing. Finally, numerical and experimental results demonstrate the success of the methodology conveyed.

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Bou-Saïd, B., G. Grau und I. Iordanoff. „On Nonlinear Rotor Dynamic Effects of Aerodynamic Bearings With Simple Flexible Rotors“. Journal of Engineering for Gas Turbines and Power 130, Nr. 1 (2008): 012503. http://dx.doi.org/10.1115/1.2747262.

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42

Zhang, Wen. „The Lower Bound of the Stability Threshold Speed of a Flexible Rotor System in Fluid-Film Bearings“. Journal of Vibration and Acoustics 111, Nr. 3 (01.07.1989): 351–53. http://dx.doi.org/10.1115/1.3269864.

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The paper is devoted to the estimation of the lower bound of the stability threshold speed (STS) of a flexible rotor system supported in fluid-film bearings. It is proved theoretically that the STS of any multi-degree-of-freedom flexible rotor system is always higher than the STS of the corresponding equivalent single disk rotor. The conclusion offers us a simple approach to estimate the STS of any actual rotor system and provides a theoretical foundation for the approach.

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Wang, Ji Yan, Yu Cheng Zhao und Chao Wang. „The Dynamic Response and Sensitivity Analysis of the SFD-Sliding Bearing Flexible Rotor System“. Advanced Materials Research 472-475 (Februar 2012): 1460–64. http://dx.doi.org/10.4028/www.scientific.net/amr.472-475.1460.

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The paper established the mechanical model of SFD-sliding bearing flexible rotor system, adopting Runge-Kutta method to solve nonlinear differential equation, thus acquiring the dynamic response and the unbalanced response curve. The study has shown: from stable periodic motion, the route of the flexible rotor system to go into chaos is: periodic motion—quasi-periodic motion—chaos—period doubling bifurcation—chaos. The paper analyzed the sensitivity of the first two critical speeds of flexible rotor system, offering design variables for optimization analysis, improving the efficiency of optimization and shortening the design cycle.

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Stanisławski, Jarosław. „Performance of Quiet Helicopter“. Transactions on Aerospace Research 2020, Nr. 1 (01.03.2020): 1–17. http://dx.doi.org/10.2478/tar-2020-0001.

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AbstractNoise generated by helicopters is one of the main problems associated with the operation of rotorcrafts. Requirements for reduction of helicopter noise were reflected in the regulations introducing lower limits of acceptable rotorcraft noise. A significant source of noise generated by helicopters are the main rotor and tail rotor blades. Radical noise reduction can be obtained by slowing down the blade tips speed of main and tail rotors. Reducing the rotational speed of the blades may decrease rotor thrust and diminish helicopter performance. The problem can be solved by attaching more blades to main rotor. The paper presents results of calculation regarding improvement of the helicopter performance which can be achieved for reduced rotor speed but with increased number of rotor blades. The calculations were performed for data of hypothetical light helicopter. Results of simulation include rotor loads and blade deformations in chosen flight conditions. Equations of motion of flexible rotor blades were solved using the Galerkin method which takes into account selected eigen modes of the blades. The simulation analyzes can help to determine the performance and loads of a quiet helicopter with reduced rotor speed within the operational envelope of helicopter flight states.

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Lesaffre, N., Jean Jacques Sinou und F. Thouverez. „Stability Analysis of a Flexible Bladed-Rotor“. Key Engineering Materials 293-294 (September 2005): 409–16. http://dx.doi.org/10.4028/www.scientific.net/kem.293-294.409.

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In any high-performance turbo-machinery, instability and damage are commonly occurring problems. The aim of this paper is to present a stability analysis of a fully-bladed flexible rotor. The flexural vibrations of the blades as well as those of the shaft are considered; the energetic approach used includes the effect of the rotational inertia. A stability detection method, bringing loci separation phenomena and coalescence, in case of an asymmetric rotor, to the fore, is made in order to determine a parametric domain where turbomachinery cannot encounter damage. Moreover, extensive parametric studies including for instance the length and the stagger angle of the blades are presented in order to obtain robust criteria for stable and unstable areas prediction. Finally, rotor/stator contact is introduced and the effect of the radial load acting on the blades when rubbing against a casing is considered.

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Lin, Yih-Hwang, und Hsiang-Chieh Yu. „Active modal control of a flexible rotor“. Mechanical Systems and Signal Processing 18, Nr. 5 (September 2004): 1117–31. http://dx.doi.org/10.1016/j.ymssp.2003.10.004.

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47

Atepor, L. „Controlling flexible rotor vibrations using parametric excitation“. Journal of Physics: Conference Series 181 (01.08.2009): 012021. http://dx.doi.org/10.1088/1742-6596/181/1/012021.

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48

Koval'chuk, P. S., und G. N. Puchka. „Biharmonic oscillation of a flexible unbalanced rotor“. International Applied Mechanics 30, Nr. 1 (Januar 1994): 58–64. http://dx.doi.org/10.1007/bf00847157.

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49

Abduljabbar, Z., M. M. ElMadany und A. A. AlAbdulwahab. „Active vibration control of a flexible rotor“. Computers & Structures 58, Nr. 3 (Februar 1996): 499–511. http://dx.doi.org/10.1016/0045-7949(95)00164-c.

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50

Lesaffre, N., J. J. Sinou und F. Thouverez. „Contact analysis of a flexible bladed-rotor“. European Journal of Mechanics - A/Solids 26, Nr. 3 (Mai 2007): 541–57. http://dx.doi.org/10.1016/j.euromechsol.2006.11.002.

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