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Active magnetic damper in a power transmission system
Commun Nonlinear Sci Numer Simulat 16 (2011) 2273–2278
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Active magnetic damper in a power transmission system D. Kozanecka *, Z. Kozanecki **, J. Łagodzin´ski *** Institute of Turbomachinery, Technical University of Lodz, 219/223 Wolczanska St., 93-005 Lodz, Poland
a r t i c l e
i n f o
Article history: Available online 23 May 2010 Keywords: Active magnetic damper Electromagnet Optimization Numerical analysis
a b s t r a c t In rotor dynamics, the bearing characteristics exerts a decisive influence on dynamics of the rotating shaft. The research and application experience have led to active magnetic bearings (AMBs), which allow for unique applications in rotating systems. The paper presents the investigations concerning optimization of the magnetic bearing construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line (aircraft industry applications) and modifies its dynamic properties. In the developed concept of AMBs for aviation purposes, a necessity of increasing its bearing load capacity and damping has occurred. The second important criterion is a weight reduction. This advanced problem leads to specific requirements on the design and materials for the AMB. To achieve these goals, some simulations have been performed. The experimental results are presented as well. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction A wide range of non-conventional applications in today’s aero industry is a serious challenge for engineers as far as specific performance and reliability issues are concerned. In order to satisfy demands imposed by a variety of complex working conditions, it is necessary to employ the latest available numerical technology allowing for optimization and simulation procedures, which help to evaluate the feasibility of application of a novel design at the early stage. Then, a prototype can be built and a series of lab tests may be conducted [2,10]. The present paper aims at showing an application of the magnetic bearing as a vibration damper in the torque transmission system to a rear rotor of the conventional helicopter [5,7]. Magnetic bearings offer a novel way of solving classical problems of rotor dynamics by suspending a spinning rotor with no contact, wear and lubrication, and controlling its dynamic behavior [1,3,4]. The paper presents the investigations concerning optimization of this magnetic support construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line, applied in the aircraft industry, and modifies its dynamic properties. 2. Concept of modification In a standard helicopter, a power transmission system is composed of several short segments of the shaft supported by rolling bearings and connected together with a series of elastic membrane couplings (Fig. 1a). Due to a significant number of supports applied at a relatively short distance along the shaft, the mentioned system is subcritical, which implies various * Corresponding author. ** Corresponding author. *** Corresponding author. E-mail addresses: [email protected] (D. Kozanecka), [email protected] (Z. Kozanecki), [email protected] (J. Łagodzin´ski). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.044
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Fig. 1. Helicopter power transmission system scheme. (a) Classical solution. (b) Novel concept with an overcritical flexible shaft supported by the active magnetic damper.
performance limitations. The solution is common in the helicopter design but its crucial disadvantage is its weight and structural complexity [2]. A concept of the proposed active vibration damping system is presented in Fig. 1b, in which a magnetic bearing has been mounted on the 5400 mm long, single-segment shaft. Thus, the power transmission system can be simplified and its weight reduced. One of the disadvantages of the system is that it has to undergo through a series of critical frequencies until the nominal power is reached during its start-up, due to a single support of the shaft and, hence, its relatively high flexibility. An application of the active magnetic bearing combined with its dynamic properties control ability allows for safe passing through those frequencies and obtaining an overcritical operation mode of the power transmission system, which is favorable from the viewpoint of operation properties [6]. A generation of the magnetic damping force by the active bearing for the particular application mentioned above is a complex issue. Apart from the control system design, it is vital to establish novel optimization procedures supporting the design of geometry of electromagnets of the bush of the bearing, a material selection of the bush and the journal, and a modeling process of the journal – bush system. It is also necessary to analyze the optimization of the design from the viewpoint of forces generated by electromagnets with simultaneous satisfaction of the minimum mass condition [5,10]. 3. Bearing models and a numerical analysis For the purpose of a numerical analysis, two geometries of the radial magnetic bearing were considered – hetero- and homopolar (Fig. 2), between which a principal difference is the loss level due to hysteresis and eddy currents. For these two geometry variants, a numerical analysis of the induction distribution in the bush and journal elements, which cooperate with a pair of electromagnets, was carried out [9] with the ANSYS Workbench tool and its Maxwell Lawsbased module – ANSYS Magnetostatic. The numerical analysis was conducted with the same core material – ARMCO iron, for which the magnetization curve B–H was established using the materials available in public domain [10]. The calculated resultant magnetic force acting on the bush at its central position was a comparative criterion for all the cases under investigation. The current values in coils and also the dimension of the radial gap in the bearing were kept constant. A series of the numerical analyses described above allowed for the electromagnet pole shape and dimension optimization. The numerical model of the bearing was then bounded by the air-propertied cuboid of the dimensions that were greater in each direction XYZ from the model by 60 mm. The Drichlet boundary condition was set on each cuboid surface. At the stage of the surface shape and pole optimization, a transverse symmetry surface of the model, perpendicular to the shaft axis, was used to set the Neuman boundary condition [9].
Fig. 2. Bearing models: heteropolar and homopolar.
D. Kozanecka et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 2273–2278
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Fig. 3. Exemplary model mesh – cores, poles, a bush and an air gap.
For each simulation, mesh generation settings (element size and type, mesh quality in the air gap) were kept constant. Depending upon the analyzed case, the model mesh was composed of 150–200 elements, with a particular care to a satisfactory number of elements in the air gaps, where the magnetic field is highly nonlinear (Fig. 3). Due to this phenomenon, using a GAP tool (Gap Aspect ratio 3:1, Gap Density: Coarse), the mesh between the poles and the bush mounted on the shaft was refined. Additionally, with a SIZING CONTROL tool, an average mesh element size in electromagnets core and the bush was set to 5 mm. The reason for that was to obtain a fine model description and, hence, more accurate results, at the expense of significantly extended computation time, however. Fig. 4 presents a numerically computed distribution of the magnetic induction for a pair of poles for both the hetero- and homopolar bearing. It has been observed that the heteropolar geometry allows for a better core material utilization. The maximum level of induction in this solution was 1.4 T, which guarantees generation of the force if the shaft is concentrically levitating in the support bush. This is a 2.4 times greater electromagnetic force if compared to the homopolar model, in which the generated force is equal to 42.5 N. In the following stage, a series of parametrical analyses for the heteropolar bearing were conducted to optimize the pole geometry as a function of the electromagnetic force generated by electromagnets. Both orthogonal and round poles were analyzed (see Section 4). In the practical realization, round poles of the diameter D = 28 mm were used. For such a concept, the generated magnetic force is equal to 122.4 N (see Fig. 5). It is the highest value that can be achieved under the assumed conditions, which, if compared to the orthogonal poles, guarantees the minimum weight of the bearing.
4. Investigation results To verify the feasibility of the modifications introduced, a helicopter tail beam test stand was prepared (Fig. 6). For the purpose of the experiment, a steel bush was mounted on the short power-transmitting shaft, which ends just behind the
Fig. 4. Magnetic induction distribution in hetero- and homopolar bearing models.
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Fig. 5. Heteropolar bearing model: magnetic force vs. pole diameter.
Fig. 6. Magnetic force measurement test stand. The a angle (unscaled) covers the whole range of the shaft movements limited by the radial clearance in the damper.
magnetic damper. This short shaft was used before at different stages of the investigations, to adjust the control system of the active bearing. At that time, a comparative criterion for both geometries (hetero- and homopolar) was a value of the currents required to elevate the shaft and its bush to the upper bounding position [8]. A significant problem to deal with was a proper interpretation of similarities between the practical experiment conditions and those assumed during the numerical simulations. In the simulations, the bush was situated ideally concentrically (e = 0) and was able to move transversely, keeping thus its axis and the magnetic damper axis parallel. In the real object, the bush performs a ‘pendulum’ movement with respect to the rotational axis crossing the center of the elastic clutch membrane connecting the power unit with the shaft. Taking into consideration a distance between the clutch membrane and the damper, and also a radial clearance between the bush and poles of the angle a = 0°30 4000 , it is concluded that the axes of the bush and the damper are practically parallel. Hence, it has been agreed that the method of force measurements in the real model is comparable with the one in the numerical simulations. In Fig. 7a electromagnets coils, which were fed during the experiment, are shown. The resultant force of the upper pair of electromagnets FXT was acting along the X axis, whereas the resultant FYT was acting along the Y axis. The total resultant force Fw was acting upwards along the vertical axis, equalizing the weight of the shaft Mw.
Fig. 7. (a) Bearing scheme (view from the free end of the shaft). (b) Real active magnetic damper test stand.
D. Kozanecka et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 2273–2278
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Fig. 8. Experimental dynamic characteristics recorded during active vibration control of the helicopter power-transmitting system prototype.
Initially during the experiment, the shaft was lying freely under its weight on the lower electromagnet poles. Next, values of the current IXT and IYT in the upper electromagnet coils were increased to the magnitudes Fw, at which the weight of the shaft with bush Mw was equalized and the shaft was elevated (Fig. 7a). During the experiment both the electromagnets in operation were fed with equal currents IXT and IYT. A comparison between the real and numerical results clearly stands for the introduced modifications. In this case, dynamic stiffness criteria, which depend both upon the electromagnets and the applied control system, were omitted. Measuring the attraction force of the electromagnets under constant base current IB = 3A and the equal radial gap, it is found that the heteropolar design elevates the shaft at IS heter = IYT = IXT =30% PWM (Pulse Width Modulation), while the homopolar design requires 50% of PWM for the control current IS homo. Assuming the relation between the force and the current in the magnetic S bearing as F w I2S , it can be found that after the substitution of the current ratio IIS homo ¼ 50%I ¼ 53 to the force ratio, a value of 30%IS S heter
25/9 = 2.77 appears. It is hence seen that for the same force required for the shaft elevation, the heteropolar design of the bearing guarantees almost a 3 times higher increase in the force. For the numerical simulation results, the force ratio is 122.4/42.5 = 2.88. This inconsistency is a result of imperfection of the numerical model, which, e.g., does not take into account an influence of the adjacent pair of electromagnets. 5. Conclusions Within the experimental scope, a series of simulations was carried out to confirm the efficiency of the optimized bearing design as an active damper in the power-transmitting system (Fig. 7b). The results indicate that there is a possibility to control vibrations of the shaft and to reduce significantly the vibration amplitude when the critical frequency passes the normal and maximum rotational speed (5000 rev/min) at the start-up. Fig. 8 presents experimentally acquired dynamic Bode characteristics of the 5400 mm long helicopter power-transmitting shaft along with damped trajectories of the shaft in the magnetic support while passing the critical frequency. The results confirm both the efficiency of the presented method and the possibility of effective vibration damping in the considered application. In spite of relatively high vibration amplitudes in the vicinity of three subsequent critical frequencies, the shaft remains within the assumed clearance safe margins and is able to reach its nominal and maximum power without any mechanical contact with the magnetic support. Within the entire operating envelope, maximum amplitudes and trajectories do not exceed 50% of the diametrical clearance of the bearing design, which was equal to 2 mm in the discussed case. The modification described in the paper provides a significant mass reduction of the system, which is crucial in any aero applications.
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References [1] Chiba A, Fukado T, Ichikawa O, Oshima M, Takemoto M, Dorrell D. Magnetic bearings and bearingless drives. Oxford, London: Elsevier; 2005. [2] DeSmidt HA. Robust – adaptive active vibration control of alloy and flexible matrix composite rotorcraft drivelines via magnetic bearings: theory and experiment. PhD dissertation. University Park, Pennsylvania, USA: Pennsylvania State University; 2005. [3] Schweitzer G, Traxler A, Bleuler H. Magnetlager. Berlin: Springer-Verlag; 1993 [in German]. [4] Schweitzer G, Maslen EH, editors. (Contributors: H. Bleuler, M. Cole, P. Keogh, R. Larsonneur, E.H. Maslen, R. Nordmann, Y. Okada, G. Schweitzer, A. Traxler.) Magnetic bearings – theory, design and application to rotating machinery. Springer-Verlag; 2009. [5] Kozanecka D. Dynamics of the flexible rotor with an additional active magnetic bearing, machine dynamics problems, vol. 25, No. 2. Warsaw, Poland; 2001. p. 21–38. [6] Kozanecka D, Kozanecki Z, Lech T. Modelling the dynamics of active magnetic bearing actuators. In: Proc world multiconference on systemics, cybernetics and informatics, SCI 2001, July 22–25, vol. IX. Industrial Parts I: USA; 2001. p. 232–235. [7] Kozanecka D, Kozanecki Z, Lech T. Theoretical and experimental investigation of dynamics of the flexible rotor with active magnetic bearings. Adv Vib Eng 2002;1(4):412–22. [8] Kozanecka D, Kozanecki Z, Lech T, Kaczmarek A. Identification of the external load of the rotating shaft supported in active magnetic bearings. In: Proc 7th conference on dynamical systems theory and applications. Łódz´, December 8–11, vol. II. 2003. p. 797–804. [9] Theory reference for ANSYS and ANSYS Workbench 11.0. ANSYS Inc.: Canonsburg, USA; 2007. Available from: www.ansys.com. [10] Łagodzin´ski J. Modeling of magnetic fields with the finite element method in machine diagnostic systems. Solid State Phenom 2009;147–149:155–60.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Active magnetic damper in a power transmission system D. Kozanecka *, Z. Kozanecki **, J. Łagodzin´ski *** Institute of Turbomachinery, Technical University of Lodz, 219/223 Wolczanska St., 93-005 Lodz, Poland
a r t i c l e
i n f o
Article history: Available online 23 May 2010 Keywords: Active magnetic damper Electromagnet Optimization Numerical analysis
a b s t r a c t In rotor dynamics, the bearing characteristics exerts a decisive influence on dynamics of the rotating shaft. The research and application experience have led to active magnetic bearings (AMBs), which allow for unique applications in rotating systems. The paper presents the investigations concerning optimization of the magnetic bearing construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line (aircraft industry applications) and modifies its dynamic properties. In the developed concept of AMBs for aviation purposes, a necessity of increasing its bearing load capacity and damping has occurred. The second important criterion is a weight reduction. This advanced problem leads to specific requirements on the design and materials for the AMB. To achieve these goals, some simulations have been performed. The experimental results are presented as well. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction A wide range of non-conventional applications in today’s aero industry is a serious challenge for engineers as far as specific performance and reliability issues are concerned. In order to satisfy demands imposed by a variety of complex working conditions, it is necessary to employ the latest available numerical technology allowing for optimization and simulation procedures, which help to evaluate the feasibility of application of a novel design at the early stage. Then, a prototype can be built and a series of lab tests may be conducted [2,10]. The present paper aims at showing an application of the magnetic bearing as a vibration damper in the torque transmission system to a rear rotor of the conventional helicopter [5,7]. Magnetic bearings offer a novel way of solving classical problems of rotor dynamics by suspending a spinning rotor with no contact, wear and lubrication, and controlling its dynamic behavior [1,3,4]. The paper presents the investigations concerning optimization of this magnetic support construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line, applied in the aircraft industry, and modifies its dynamic properties. 2. Concept of modification In a standard helicopter, a power transmission system is composed of several short segments of the shaft supported by rolling bearings and connected together with a series of elastic membrane couplings (Fig. 1a). Due to a significant number of supports applied at a relatively short distance along the shaft, the mentioned system is subcritical, which implies various * Corresponding author. ** Corresponding author. *** Corresponding author. E-mail addresses: [email protected] (D. Kozanecka), [email protected] (Z. Kozanecki), [email protected] (J. Łagodzin´ski). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.044
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Fig. 1. Helicopter power transmission system scheme. (a) Classical solution. (b) Novel concept with an overcritical flexible shaft supported by the active magnetic damper.
performance limitations. The solution is common in the helicopter design but its crucial disadvantage is its weight and structural complexity [2]. A concept of the proposed active vibration damping system is presented in Fig. 1b, in which a magnetic bearing has been mounted on the 5400 mm long, single-segment shaft. Thus, the power transmission system can be simplified and its weight reduced. One of the disadvantages of the system is that it has to undergo through a series of critical frequencies until the nominal power is reached during its start-up, due to a single support of the shaft and, hence, its relatively high flexibility. An application of the active magnetic bearing combined with its dynamic properties control ability allows for safe passing through those frequencies and obtaining an overcritical operation mode of the power transmission system, which is favorable from the viewpoint of operation properties [6]. A generation of the magnetic damping force by the active bearing for the particular application mentioned above is a complex issue. Apart from the control system design, it is vital to establish novel optimization procedures supporting the design of geometry of electromagnets of the bush of the bearing, a material selection of the bush and the journal, and a modeling process of the journal – bush system. It is also necessary to analyze the optimization of the design from the viewpoint of forces generated by electromagnets with simultaneous satisfaction of the minimum mass condition [5,10]. 3. Bearing models and a numerical analysis For the purpose of a numerical analysis, two geometries of the radial magnetic bearing were considered – hetero- and homopolar (Fig. 2), between which a principal difference is the loss level due to hysteresis and eddy currents. For these two geometry variants, a numerical analysis of the induction distribution in the bush and journal elements, which cooperate with a pair of electromagnets, was carried out [9] with the ANSYS Workbench tool and its Maxwell Lawsbased module – ANSYS Magnetostatic. The numerical analysis was conducted with the same core material – ARMCO iron, for which the magnetization curve B–H was established using the materials available in public domain [10]. The calculated resultant magnetic force acting on the bush at its central position was a comparative criterion for all the cases under investigation. The current values in coils and also the dimension of the radial gap in the bearing were kept constant. A series of the numerical analyses described above allowed for the electromagnet pole shape and dimension optimization. The numerical model of the bearing was then bounded by the air-propertied cuboid of the dimensions that were greater in each direction XYZ from the model by 60 mm. The Drichlet boundary condition was set on each cuboid surface. At the stage of the surface shape and pole optimization, a transverse symmetry surface of the model, perpendicular to the shaft axis, was used to set the Neuman boundary condition [9].
Fig. 2. Bearing models: heteropolar and homopolar.
D. Kozanecka et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 2273–2278
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Fig. 3. Exemplary model mesh – cores, poles, a bush and an air gap.
For each simulation, mesh generation settings (element size and type, mesh quality in the air gap) were kept constant. Depending upon the analyzed case, the model mesh was composed of 150–200 elements, with a particular care to a satisfactory number of elements in the air gaps, where the magnetic field is highly nonlinear (Fig. 3). Due to this phenomenon, using a GAP tool (Gap Aspect ratio 3:1, Gap Density: Coarse), the mesh between the poles and the bush mounted on the shaft was refined. Additionally, with a SIZING CONTROL tool, an average mesh element size in electromagnets core and the bush was set to 5 mm. The reason for that was to obtain a fine model description and, hence, more accurate results, at the expense of significantly extended computation time, however. Fig. 4 presents a numerically computed distribution of the magnetic induction for a pair of poles for both the hetero- and homopolar bearing. It has been observed that the heteropolar geometry allows for a better core material utilization. The maximum level of induction in this solution was 1.4 T, which guarantees generation of the force if the shaft is concentrically levitating in the support bush. This is a 2.4 times greater electromagnetic force if compared to the homopolar model, in which the generated force is equal to 42.5 N. In the following stage, a series of parametrical analyses for the heteropolar bearing were conducted to optimize the pole geometry as a function of the electromagnetic force generated by electromagnets. Both orthogonal and round poles were analyzed (see Section 4). In the practical realization, round poles of the diameter D = 28 mm were used. For such a concept, the generated magnetic force is equal to 122.4 N (see Fig. 5). It is the highest value that can be achieved under the assumed conditions, which, if compared to the orthogonal poles, guarantees the minimum weight of the bearing.
4. Investigation results To verify the feasibility of the modifications introduced, a helicopter tail beam test stand was prepared (Fig. 6). For the purpose of the experiment, a steel bush was mounted on the short power-transmitting shaft, which ends just behind the
Fig. 4. Magnetic induction distribution in hetero- and homopolar bearing models.
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Fig. 5. Heteropolar bearing model: magnetic force vs. pole diameter.
Fig. 6. Magnetic force measurement test stand. The a angle (unscaled) covers the whole range of the shaft movements limited by the radial clearance in the damper.
magnetic damper. This short shaft was used before at different stages of the investigations, to adjust the control system of the active bearing. At that time, a comparative criterion for both geometries (hetero- and homopolar) was a value of the currents required to elevate the shaft and its bush to the upper bounding position [8]. A significant problem to deal with was a proper interpretation of similarities between the practical experiment conditions and those assumed during the numerical simulations. In the simulations, the bush was situated ideally concentrically (e = 0) and was able to move transversely, keeping thus its axis and the magnetic damper axis parallel. In the real object, the bush performs a ‘pendulum’ movement with respect to the rotational axis crossing the center of the elastic clutch membrane connecting the power unit with the shaft. Taking into consideration a distance between the clutch membrane and the damper, and also a radial clearance between the bush and poles of the angle a = 0°30 4000 , it is concluded that the axes of the bush and the damper are practically parallel. Hence, it has been agreed that the method of force measurements in the real model is comparable with the one in the numerical simulations. In Fig. 7a electromagnets coils, which were fed during the experiment, are shown. The resultant force of the upper pair of electromagnets FXT was acting along the X axis, whereas the resultant FYT was acting along the Y axis. The total resultant force Fw was acting upwards along the vertical axis, equalizing the weight of the shaft Mw.
Fig. 7. (a) Bearing scheme (view from the free end of the shaft). (b) Real active magnetic damper test stand.
D. Kozanecka et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 2273–2278
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Fig. 8. Experimental dynamic characteristics recorded during active vibration control of the helicopter power-transmitting system prototype.
Initially during the experiment, the shaft was lying freely under its weight on the lower electromagnet poles. Next, values of the current IXT and IYT in the upper electromagnet coils were increased to the magnitudes Fw, at which the weight of the shaft with bush Mw was equalized and the shaft was elevated (Fig. 7a). During the experiment both the electromagnets in operation were fed with equal currents IXT and IYT. A comparison between the real and numerical results clearly stands for the introduced modifications. In this case, dynamic stiffness criteria, which depend both upon the electromagnets and the applied control system, were omitted. Measuring the attraction force of the electromagnets under constant base current IB = 3A and the equal radial gap, it is found that the heteropolar design elevates the shaft at IS heter = IYT = IXT =30% PWM (Pulse Width Modulation), while the homopolar design requires 50% of PWM for the control current IS homo. Assuming the relation between the force and the current in the magnetic S bearing as F w I2S , it can be found that after the substitution of the current ratio IIS homo ¼ 50%I ¼ 53 to the force ratio, a value of 30%IS S heter
25/9 = 2.77 appears. It is hence seen that for the same force required for the shaft elevation, the heteropolar design of the bearing guarantees almost a 3 times higher increase in the force. For the numerical simulation results, the force ratio is 122.4/42.5 = 2.88. This inconsistency is a result of imperfection of the numerical model, which, e.g., does not take into account an influence of the adjacent pair of electromagnets. 5. Conclusions Within the experimental scope, a series of simulations was carried out to confirm the efficiency of the optimized bearing design as an active damper in the power-transmitting system (Fig. 7b). The results indicate that there is a possibility to control vibrations of the shaft and to reduce significantly the vibration amplitude when the critical frequency passes the normal and maximum rotational speed (5000 rev/min) at the start-up. Fig. 8 presents experimentally acquired dynamic Bode characteristics of the 5400 mm long helicopter power-transmitting shaft along with damped trajectories of the shaft in the magnetic support while passing the critical frequency. The results confirm both the efficiency of the presented method and the possibility of effective vibration damping in the considered application. In spite of relatively high vibration amplitudes in the vicinity of three subsequent critical frequencies, the shaft remains within the assumed clearance safe margins and is able to reach its nominal and maximum power without any mechanical contact with the magnetic support. Within the entire operating envelope, maximum amplitudes and trajectories do not exceed 50% of the diametrical clearance of the bearing design, which was equal to 2 mm in the discussed case. The modification described in the paper provides a significant mass reduction of the system, which is crucial in any aero applications.
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References [1] Chiba A, Fukado T, Ichikawa O, Oshima M, Takemoto M, Dorrell D. Magnetic bearings and bearingless drives. Oxford, London: Elsevier; 2005. [2] DeSmidt HA. Robust – adaptive active vibration control of alloy and flexible matrix composite rotorcraft drivelines via magnetic bearings: theory and experiment. PhD dissertation. University Park, Pennsylvania, USA: Pennsylvania State University; 2005. [3] Schweitzer G, Traxler A, Bleuler H. Magnetlager. Berlin: Springer-Verlag; 1993 [in German]. [4] Schweitzer G, Maslen EH, editors. (Contributors: H. Bleuler, M. Cole, P. Keogh, R. Larsonneur, E.H. Maslen, R. Nordmann, Y. Okada, G. Schweitzer, A. Traxler.) Magnetic bearings – theory, design and application to rotating machinery. Springer-Verlag; 2009. [5] Kozanecka D. Dynamics of the flexible rotor with an additional active magnetic bearing, machine dynamics problems, vol. 25, No. 2. Warsaw, Poland; 2001. p. 21–38. [6] Kozanecka D, Kozanecki Z, Lech T. Modelling the dynamics of active magnetic bearing actuators. In: Proc world multiconference on systemics, cybernetics and informatics, SCI 2001, July 22–25, vol. IX. Industrial Parts I: USA; 2001. p. 232–235. [7] Kozanecka D, Kozanecki Z, Lech T. Theoretical and experimental investigation of dynamics of the flexible rotor with active magnetic bearings. Adv Vib Eng 2002;1(4):412–22. [8] Kozanecka D, Kozanecki Z, Lech T, Kaczmarek A. Identification of the external load of the rotating shaft supported in active magnetic bearings. In: Proc 7th conference on dynamical systems theory and applications. Łódz´, December 8–11, vol. II. 2003. p. 797–804. [9] Theory reference for ANSYS and ANSYS Workbench 11.0. ANSYS Inc.: Canonsburg, USA; 2007. Available from: www.ansys.com. [10] Łagodzin´ski J. Modeling of magnetic fields with the finite element method in machine diagnostic systems. Solid State Phenom 2009;147–149:155–60.