**1. Introduction**

presents a study on the rigid, semi-rigid and flexible beam-to-beam connections effect and the influence of steel-concrete interaction degree over the non-linear dynamic behavior of composite floors when subjected to human rhythmic activities. Chapter 12 deals with the problem of dynamic modelling and parametric vibration of transmission mechanisms with elastic components governed by linearized differential equations having time-varying coefficients. Numerical procedures based on Runge-Kutta and Newmark methods are proposed and applied to find periodic solutions of linear differential equations with timeperiodic coefficients. Chapter 13 describes an approach to observe the vibrations of a low Earth orbiting satellite's solar array paddle induced by thermal shock using an onboard CMOS camera mounted on a Greenhouse gases Observing Satellite launched by the Japan Aerospace Exploration Agency (JAXA) in 2009. Chapter 14 concludes the book describing an experimental study on optimal vibrotactile stimulation to activate the parasympathetic

Finally, I would like to express my sincere gratitude to all the authors for their excellent contributions, which I am sure will be valuable to the readers. I would also like to thank the editorial staff of InTech for their great effort and support in the process of edition and

I truly hope that this book can be useful and inspiring for contributing to the technology development, new academic and industrial research and many inventions and innovations

> **Francisco Beltrán Carbajal** Departamento de Energía

> > Mexico

Universidad Autónoma Metropolitana, Unidad Azcapotzalco

nerve system.

VIII Preface

publication of the book.

in Vibration Engineering and Structural Dynamics.

In the last decades the deeper and more detailed understanding of rotating machinery dynam‐ ic behavior facilitated the study and the design of several devices aiming at friction reduction, vibration damping and control, rotational speed increase and mechanical design optimization. Among these devices a promising technology is represented by magnetic actuators used as bearings which found a great spread in rotordynamics and in high precision applications. A first classification of magnetic bearings according to the physical working principle allows to pick out two main families: a) Active Magnetic Bearings [1], [2], making use of an electronic control unit to regulate the current flowing in the coils of the actuators. They need external source of energy. b) Passive Magnetic Bearings [3], [4], [5]: they do not need any electronic equipment. The control of the mechanical structure is achieved without the introduction of any external energy source. They exploit the reluctance force or the Lorentz force due to the gener‐ ation of eddy currents developed in a conductor in a relative motion in a magnetic field.Active Magnetic Bearings require sensors an electronic equipment but, although more expensive re‐ spect to classical ball bearings, they offer several technological advantages:


© 2012 Tonoli et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Tonoli et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**•** Diagnostics are readily performed, as the states of the rotor are measured for the opera‐ tion of the AMB anyway, and this information can be used to check operating conditions and performance. Even active diagnostics are feasible, by using the AMB as actuators for

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**•** The lower maintenance costs and higher life time of an AMB have been demonstrated un‐ der severe conditions. Essentially, they are due to the lack of mechanical wear. Currently, this is the main reason for the increasing number of applications in turbomachinery;

**•** The cost structure of an AMB is that of a typical mechatronics product. The costs for de‐ veloping a prototype, mainly because of the demanding software, can be rather high. On the other side, a series production will lower the costs considerably because of the porta‐

Active Magnetic Bearings can be classified as a typical mechatronic product due to its nature which involves mechanical, electrical and control aspects, merging them in a single system. Rotordynamic field offer several examples of application areas [1], [6] : (a) Turbomachinery, (b) Vibration isolation, (c) Machine tools and electric drives, (d) Energy storing flywheels, (e) Instruments in space and physics, (f) Non-contacting suspensions for micro-techniques, (g) Identification and test equipment in rotordynamics, (f) Microapplications such as gyroscopic sensors [7], [8].The attractive potential of active magnetic suspensions motivated a consider‐ able research effort for the past decade focused mostly on electrical actuation subsystem and

This chapter illustrates the design, the modeling, the experimental tests and validation of all subsystems of a rotor on a five-axes active magnetic suspension. The mechanical, elec‐ trical, electronic and control strategies aspects are explained with a mechatronic approach evaluating all the interactions between them. The main goals of the manuscript are: a) Il‐ lustrate the design and the modeling phases of a five-axes active magnetic suspension; b) Discuss the design steps and the practical implementation of a standard suspension con‐ trol strategy; c) Introduce an off-line technique of electrical centering of the actuators. The experimental test rig is a shaft (Weight: 5.3 kg. Length: 0.5 m) supported by two radial and one axial cylindrical active magnetic bearings and powered by an asynchronous high frequency electric motor.The chapter starts on an overview of the most common technolo‐ gies used to support rotors with a deep analysis of their advantages and drawbacks with respect to active magnetic bearings. Furthermore a discussion on magnetic suspensions state of the art is carried out highlighting the research efforts directions and the goals reached in the last years.In the central sections, a detailed description of each subsystem is performed along with the modeling steps. In particular the rotor is modeled with a FE

The last sections of the chapter are focused on the control strategies design and the experi‐ mental tests.An off-line technique of actuators electrical centering is explained and its ad‐ vantages are described in the control design context. This strategy can be summarized as follows. Knowing that: a) each actuation axis is composed by two electromagnets; b) each electromagnet needs a current closed-loop control; c) the bandwidth of this control is de‐ pending on the mechanical Airgap,then the technique allows obtaining the same value of

generating well defined test signals simultaneously with their bearing function;

bility of that software.

control strategies [3], [9], [10], [11], [12], [13], [14].

code while the actuators are considered in a linearized model.


**•** Diagnostics are readily performed, as the states of the rotor are measured for the opera‐ tion of the AMB anyway, and this information can be used to check operating conditions and performance. Even active diagnostics are feasible, by using the AMB as actuators for generating well defined test signals simultaneously with their bearing function;

**•** Viscous friction can be avoided if the rotor is confined in high vacuum;

**•** Achievable fast positioning and/or high rotational speed of the rotor;

**•** Further statements about the technology of realization can be done:

**•** The small sensitivity to the operating conditions;

2 Advances in Vibration Engineering and Structural Dynamics

**•** The predictability of the behavior.

tions still needs special attention;

becomes much larger;

**•** Dynamics adaptable to the desired application by tuning of the control loop;

**•** Precise positioning of the rotor due to the control loop: this is mainly determined by the quality of the measurement signal within the control loop. Conventional inductive sen‐ sors, for example, have a measurement resolution of about 1 ÷ 1000μm of a millimeter;

**•** The gap between rotor and bearing amounts typically to a few tenths of a millimeter, but for specific applications it can be as large as 20 mm. In that case, of course, the bearing

**•** The rotor can be allowed to rotate at high speeds. The high circumferential speed in the bearing, only limited by the strength of material of the rotor, offers the possibilities of de‐ signing new machines with higher power density and of realizing novel constructions. Actually, about 350 m/s are achievable, for example by using amorphous metals which can sustain high stresses and at the same time have very good soft-magnetic properties, or by binding the rotor laminations with carbon fibers. Design advantages result from the absence of lubrication seals and from the possibility of having a higher shaft diameter at

**•** The specific load capacity of the bearing depends on the type of ferromagnetic material and the design of the bearing electromagnet. It will be about 20 N/cm2 and can be as high as 40 N/cm2. The reference area is the cross sectional area of the bearing. Thus the maxi‐

**•** The bearing and the rotor can be integrated on the same shaft by realizing bearingless config‐ urations which allow to reduce the size of the system and to perform a cost saving solution.

**•** Retainer bearings are additional ball or journal bearings, which in normal operation are not in contact with the rotor. In case of overload or malfunction of the AMB they have to operate for a very short time: they keep the spinning rotor from touching the housing un‐ til the rotor comes to rest or until the AMB regains control of the rotor. The design of such retainer bearings depends on the specific application and despite a variety of good solu‐

**•** The unbalance compensation and the force-free rotation are control features where the vi‐ brations due to residual unbalance are measured and identified by the AMB. The signal is used to either generate counteracting and compensating bearing forces or to shift the ro‐

the bearing site. This makes the shaft stiffer and less sensitive to vibrations;

mum bearing load is mainly a function of the bearing size;

tor axis in such a way that the rotor is rotating force-free;

**•** Low vibration level;


Active Magnetic Bearings can be classified as a typical mechatronic product due to its nature which involves mechanical, electrical and control aspects, merging them in a single system. Rotordynamic field offer several examples of application areas [1], [6] : (a) Turbomachinery, (b) Vibration isolation, (c) Machine tools and electric drives, (d) Energy storing flywheels, (e) Instruments in space and physics, (f) Non-contacting suspensions for micro-techniques, (g) Identification and test equipment in rotordynamics, (f) Microapplications such as gyroscopic sensors [7], [8].The attractive potential of active magnetic suspensions motivated a consider‐ able research effort for the past decade focused mostly on electrical actuation subsystem and control strategies [3], [9], [10], [11], [12], [13], [14].

This chapter illustrates the design, the modeling, the experimental tests and validation of all subsystems of a rotor on a five-axes active magnetic suspension. The mechanical, elec‐ trical, electronic and control strategies aspects are explained with a mechatronic approach evaluating all the interactions between them. The main goals of the manuscript are: a) Il‐ lustrate the design and the modeling phases of a five-axes active magnetic suspension; b) Discuss the design steps and the practical implementation of a standard suspension con‐ trol strategy; c) Introduce an off-line technique of electrical centering of the actuators. The experimental test rig is a shaft (Weight: 5.3 kg. Length: 0.5 m) supported by two radial and one axial cylindrical active magnetic bearings and powered by an asynchronous high frequency electric motor.The chapter starts on an overview of the most common technolo‐ gies used to support rotors with a deep analysis of their advantages and drawbacks with respect to active magnetic bearings. Furthermore a discussion on magnetic suspensions state of the art is carried out highlighting the research efforts directions and the goals reached in the last years.In the central sections, a detailed description of each subsystem is performed along with the modeling steps. In particular the rotor is modeled with a FE code while the actuators are considered in a linearized model.

The last sections of the chapter are focused on the control strategies design and the experi‐ mental tests.An off-line technique of actuators electrical centering is explained and its ad‐ vantages are described in the control design context. This strategy can be summarized as follows. Knowing that: a) each actuation axis is composed by two electromagnets; b) each electromagnet needs a current closed-loop control; c) the bandwidth of this control is de‐ pending on the mechanical Airgap,then the technique allows obtaining the same value of the closed-loop bandwidth of the current control of both the electromagnets on the same ac‐ tuation axis. This approach improves performance and gives more steadiness to the control behavior.The decentralized approach of the control strategy allowing the full suspensions on five axes is illustrated from the design steps to the practical implementation on the con‐ trol unit.Finally, the experimental tests are carried out on the rotor to validate the suspen‐ sion control and the off-line electrical centering. The numerical and experimental results are superimposed and compared to prove the effectiveness of the modeling approach.

Table 1 reports the main parameters of the rotor. Figure 2 illustrates the section view of the

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Active Magnetic Bearings applied to rotating machines can be considered as a typical me‐ chatronic application, since it involves the control of mechanical system (the rotor) by means of an electronic control unit which elaborates the commands to feed electrical power drivers regulating the electromechanical actuators. The information to perform closed loop control

**Figure 2.** System section view: 1) tang, 2) radial sensor, 3) radial AMB support, 4) radial AMB laminated stator, 5) axial AMB disc, 6) radial AMB laminated stator, 7) axial sensor, 8) sensor cap, 9) bushing cap, 10) axial AMB electromagnet,

The interactions and the main functions of these subsystems are highlighted in Figure 3.The reported scheme is a standard representation of the system. However each block can be of different nature depending on the application. A short summary of the technologies typical‐ ly used for each subsystem and the technologies used in the rig described in this chapter are

Two main families of control architecture can be listed for active magnetic bearings:

**•** Decentralized SISO control: the action of each actuator is independent from the others

**•** Centralized MIMO control: actuators are coupled as well as sensors information. A single

Several control strategies have been implemented and tested on rotors equipped with active

11) electric motor, 12) motor support, 13) foundation, 14) threaded ring, 15) sensor cap.

and exploits a dedicated control law and sensors information;

control action is devoted to feed power drivers.

reported in the following sections.

**2.1. Control**

magnetic bearings:

**•** Gain scheduled control [15];

**•** Adaptive control [16], [17];

system showing the layout of sensors, actuators, motor and rotor.

architecture is given by displacement and current sensors.

## **2. System Architecture**

The rig used for the modeling, the design and the experimental tests is an electrical spindle (picture reported in Figure 1) consisting of a shaft supported by two radial and one axial ac‐ tive magnetic bearings with cylindrical geometry and powered by an asynchronous high frequency electric motor. Two mechanical ball bearings, with radial and axial airgaps equal to half of the levitation ones, are positioned at the ends of the shaft to guarantee a safely touch-down of the shaft for anomalous working conditions with excessive whirling ampli‐ tude. The rotation axis is horizontal and the weight has the direction of each bearing.

**Figure 1.** Picture of the rotor.


**Table 1.** Rotor mechanical and geometric parameters.

Table 1 reports the main parameters of the rotor. Figure 2 illustrates the section view of the system showing the layout of sensors, actuators, motor and rotor.

Active Magnetic Bearings applied to rotating machines can be considered as a typical me‐ chatronic application, since it involves the control of mechanical system (the rotor) by means of an electronic control unit which elaborates the commands to feed electrical power drivers regulating the electromechanical actuators. The information to perform closed loop control architecture is given by displacement and current sensors.

**Figure 2.** System section view: 1) tang, 2) radial sensor, 3) radial AMB support, 4) radial AMB laminated stator, 5) axial AMB disc, 6) radial AMB laminated stator, 7) axial sensor, 8) sensor cap, 9) bushing cap, 10) axial AMB electromagnet, 11) electric motor, 12) motor support, 13) foundation, 14) threaded ring, 15) sensor cap.

The interactions and the main functions of these subsystems are highlighted in Figure 3.The reported scheme is a standard representation of the system. However each block can be of different nature depending on the application. A short summary of the technologies typical‐ ly used for each subsystem and the technologies used in the rig described in this chapter are reported in the following sections.

## **2.1. Control**

the closed-loop bandwidth of the current control of both the electromagnets on the same ac‐ tuation axis. This approach improves performance and gives more steadiness to the control behavior.The decentralized approach of the control strategy allowing the full suspensions on five axes is illustrated from the design steps to the practical implementation on the con‐ trol unit.Finally, the experimental tests are carried out on the rotor to validate the suspen‐ sion control and the off-line electrical centering. The numerical and experimental results are

The rig used for the modeling, the design and the experimental tests is an electrical spindle (picture reported in Figure 1) consisting of a shaft supported by two radial and one axial ac‐ tive magnetic bearings with cylindrical geometry and powered by an asynchronous high frequency electric motor. Two mechanical ball bearings, with radial and axial airgaps equal to half of the levitation ones, are positioned at the ends of the shaft to guarantee a safely touch-down of the shaft for anomalous working conditions with excessive whirling ampli‐

tude. The rotation axis is horizontal and the weight has the direction of each bearing.

**Parameter Symbol Value Unit** Rotor mass *m* 5.31 kg Rotor transversal Inertia *Jx* = *Jy* 1.153∙10-1 Kg/m2 Rotor polar Inertia *Jz* 1.826∙10-3 Kg/m2 Bearing rad. 1 location *a* 214.5 mm A/V Bearing rad. 2 location *b* 212.6 mm A/V Axial/Radial Airgap *g* 0.75e-3 mm Isotropic support stiffness *f* / *x* 2.5∙10-5 N/m

superimposed and compared to prove the effectiveness of the modeling approach.

**2. System Architecture**

4 Advances in Vibration Engineering and Structural Dynamics

**Figure 1.** Picture of the rotor.

**Table 1.** Rotor mechanical and geometric parameters.

Two main families of control architecture can be listed for active magnetic bearings:


Several control strategies have been implemented and tested on rotors equipped with active magnetic bearings:


**2.2. Power drivers**

The electronic circuits of power amplification stage to convert low power controller output signal to a high power stator input signal is chosen according to the kind of application. Ba‐

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**1.** Linear analogue amplifiers have push–pull transistors at the output stage. They allow to enhance the current capability and to integrate a high-gain linear amplifier, such as a power operational amplifier. The linear amplifiers have the advantage of precise cur‐ rent and voltage regulation as well as low noise and they have a current rating of less than 10 A. Operation at the rated current is available only with effective cooling with heat sinks. Therefore the amplifier dimensions are large, resulting in high cost. The effi‐

**2.** Switched-mode amplifiers enhance efficiency. Since the losses in the power devices are re‐ duced, the heat sinks are much smaller and, as a result, switched mode amplifiers are com‐ pact in dimension so that the cost is low. Switched-mode operation of power devices is widely used in industry, e.g., for general-purpose inverters in ac drives and computer power supplies. This category of amplifiers is dominant in magnetic bearing drivers. Hybrid amplifiers take advantage of linear and switched-mode amplifiers. At low current the push–pull transistors operate as a linear amplifier but at high current they operate in switched mode. To take advantage of a hybrid amplifier, it is quite important to modify the

The rig object of study in this chapter is equipped with an H-bridge switching amplifier for each actuator.The power stage consists of an Embedded Isolated Power Module Board with four fully independent MOSFET/IGBT legs that supports up to 25 amperes with 100 volt of DC Bus. Also a maximum PWM switching frequency is 80kHz making this module suitable for high performance driving applications where control loop bandwidth and current ripple

are important factors.The scheme used to feed power drivers is reported in Figure 4.

sically, three main families of electronic circuits can be identified:

ciency is low because of high losses in the push–pull transistor.

winding structure in magnetic bearings.

**Figure 4.** Power driver scheme.


**Figure 3.** Overall system architecture.

In this chapter a decentralized PID strategy is implemented on a control module equip‐ ped with a DSP/FPGA–based digital control unit (EKU2.1). This digital platform allows the rapid reconfigurations of the overall system throw up to 108 (from FPGA) and 46 (from DSP) configurable digital I/O lines for input/output, event, PWM, capture/genera‐ tion and user functions.Both DSP and FPGA have a dedicated Hard Real-Time Operating System (HRTOS) based on a non-pre-emptive scheduler (DSP side), involving ISR time or event triggered. EKU2.1 uses a single-master (DSP) multi-slave (FPGA) point-to-point communication protocol, based on Wishbone format; a system bus manages data ex‐ change between the two cores. Software code is developed using the target-dependent tools Texas Instruments® Code Composer Studio.

## **2.2. Power drivers**

**•** Robust H∞ control [18];

**•** Optimal control [21];

**•** Robust sliding mode control [19];

6 Advances in Vibration Engineering and Structural Dynamics

**•** Dynamic programming control [22];

**•** Feedback linearization control [25];

**•** Control by transfer function approach [27];

**•** Genetic algorithm control [23];

**•** Fuzzy logic control [24];

**•** Time-delay control [26];

**•** μ-synthesis control [28].

**Figure 3.** Overall system architecture.

tools Texas Instruments® Code Composer Studio.

**•** Robust control via eigenstructure assignment dynamical compensation [20];

In this chapter a decentralized PID strategy is implemented on a control module equip‐ ped with a DSP/FPGA–based digital control unit (EKU2.1). This digital platform allows the rapid reconfigurations of the overall system throw up to 108 (from FPGA) and 46 (from DSP) configurable digital I/O lines for input/output, event, PWM, capture/genera‐ tion and user functions.Both DSP and FPGA have a dedicated Hard Real-Time Operating System (HRTOS) based on a non-pre-emptive scheduler (DSP side), involving ISR time or event triggered. EKU2.1 uses a single-master (DSP) multi-slave (FPGA) point-to-point communication protocol, based on Wishbone format; a system bus manages data ex‐ change between the two cores. Software code is developed using the target-dependent The electronic circuits of power amplification stage to convert low power controller output signal to a high power stator input signal is chosen according to the kind of application. Ba‐ sically, three main families of electronic circuits can be identified:


Hybrid amplifiers take advantage of linear and switched-mode amplifiers. At low current the push–pull transistors operate as a linear amplifier but at high current they operate in switched mode. To take advantage of a hybrid amplifier, it is quite important to modify the winding structure in magnetic bearings.

The rig object of study in this chapter is equipped with an H-bridge switching amplifier for each actuator.The power stage consists of an Embedded Isolated Power Module Board with four fully independent MOSFET/IGBT legs that supports up to 25 amperes with 100 volt of DC Bus. Also a maximum PWM switching frequency is 80kHz making this module suitable for high performance driving applications where control loop bandwidth and current ripple are important factors.The scheme used to feed power drivers is reported in Figure 4.

**Figure 4.** Power driver scheme.

Standard AMBs equipment for rotors suspensions are realized with five couples of cylindri‐ cal shape electromagnets to perform five dof active control. Conical shape of magnetic bear‐ ings exerting forces both in axial and radial direction simultaneously allow to save one couple of electromagnets and hence to reduce the size although the bearing design results more complex than standard cylindrical solution. This geometry permits to reachhigher ro‐ tation speed, limited in cylindrical solution by the strains growing in axial bearing disc.

**Parameter Symbol Value Unit**

Number of turns *NAX* 120 - Circuitation length *lAX* 48e-3 m Active section on airgap *SAX* 1210e-6 m2 Nominal airgap *g*0*AX* 0.75e-3 m Resistance *RAX* 0.5 Ω Nominal inductance *L* 0*AX* 0.0146 H

AXIAL Actuator

RADIAL actuator

An important part of the performance of a magnetic bearing depends on the characteristics of the displacement sensors used. In order to measure the position of a moving rotor, con‐ tact-free sensors must be used which, moreover, must be able to measure on a rotating sur‐

*RAD* 135.2e-3 m

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Number of turns *NRAD* 110 -

Active section on airgap *SRAD* 480e-6 m2 Nominalairgap *g*0*RAD* 0.75e-3 M Resistance *RRAD* 0.5 Ω Nominal Inductance *L* 0*RAD* 0.0049 H

Circuitation length *l*

**Table 2.** Actuators parameters.

**Figure 6.** Actuators configuration.

**2.4. Sensors**

**Figure 5.** Geometry of actuation stage. a) Conical profile. b) Cylindrical profile.

### **2.3. Actuators**

Actuators geometry and configuration depends on the electromagnets profile and on the number of actuators per actuation stage.

In this work classical configuration with four cylindrical actuators per actuation stage is dealt with. The disposition of the ten electromagnets is illustrated in Figure 6.

The main electrical and geometrical actuators parameters are listed in Table 2.



**Table 2.** Actuators parameters.

Standard AMBs equipment for rotors suspensions are realized with five couples of cylindri‐ cal shape electromagnets to perform five dof active control. Conical shape of magnetic bear‐ ings exerting forces both in axial and radial direction simultaneously allow to save one couple of electromagnets and hence to reduce the size although the bearing design results more complex than standard cylindrical solution. This geometry permits to reachhigher ro‐ tation speed, limited in cylindrical solution by the strains growing in axial bearing disc.

**Figure 5.** Geometry of actuation stage. a) Conical profile. b) Cylindrical profile.

number of actuators per actuation stage.

8 Advances in Vibration Engineering and Structural Dynamics

Actuators geometry and configuration depends on the electromagnets profile and on the

In this work classical configuration with four cylindrical actuators per actuation stage is

dealt with. The disposition of the ten electromagnets is illustrated in Figure 6.

The main electrical and geometrical actuators parameters are listed in Table 2.

**Parameter Symbol Value Unit** Vacuum permeability μ*<sup>0</sup>* 1.26e-006 H/m Voltage supply *V DC* 50 V

**2.3. Actuators**

**Figure 6.** Actuators configuration.

#### **2.4. Sensors**

An important part of the performance of a magnetic bearing depends on the characteristics of the displacement sensors used. In order to measure the position of a moving rotor, con‐ tact-free sensors must be used which, moreover, must be able to measure on a rotating sur‐ face. Consequently, the geometry of the rotor, i.e. its surface quality, and the homogeneity of the material at the sensor will also influence the measuring results. A bad surface will thus produce noise disturbances, and geometry errors may cause disturbances with the rotational frequency or with multiples thereof.

Many modeling techniques can be adopted; an analytical rigid approach (based on the 4 d.o.f. rotor modeling) is here presented beside the most common Finite Element (FE) ap‐

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The main hypothesis here adopted is to consider a constant spin speed. In this case the rotor behavior on the X-Y plane (known as flexural) is not coupled with the behavior on the Z-direction (axial). Other important assumption is that any rotation (except for spin‐

A simple description of the rotor model block diagram is presented in Figure 8. Forces (due to AMBs, motor and external) acting on the rotor are un-grouped for the X-Y and for the Z behavior, these signals are fed to the block describing the dynamic behaviors. Outputs of these blocks are the states (displacements and velocities) of the systems, re‐ ported as displacements and speeds to sensors, AMBs and motor. The constant spin speed *Ω* is used in the X-Y model for the gyroscopic behavior and reported as out‐ put.Spin speed and displacement on the sensors are physical entities measured by specific sensors, and signal are reported to the Sensor block; the other displacements and relative velocities (to AMBs and motor) should be used for intrinsic feedback such as back electro‐

Model inputs are the forces acting on the rotor, while outputs are typically displacements either on sensors or on AMBs and motor (Figure 9). The rotor is suspended by two radial magnetic bearing (AMB1 and AMB2) which generate four forces oriented as the reference plant reference frame and acting in the center of the relative AMB; these forces act the be‐ havior on X-Y plane. A further magnetic bearing (AMB3) is used to constrain displacements along Z axis (axial). The five forces due to AMBs are collected in vector fAMB. The electric

proach. The discretization for FEM software is reported in Figure 7.b.

**Figure 7.** Rotor section view. a) View with dimensions. b) Discretization for FEM modeling.

ning rotation) should be small.

motive force in the motor or magnetic bearings.

**3.1. Model Block Diagram**

*3.1.1. Model inputs / outputs*

In addition, depending on the application, speeds, currents, flux densities and temperatures are to be measured in magnetic bearing systems.

When selecting the displacement sensors, depending on the application of the magnetic bearing, measuring range, linearity, sensitivity, resolution, and frequency range are to be taken into account as well as:


The most important displacement sensors technologies are:


The rig described in this chapter is equipped with five eddy current displacement sensors: high-frequency alternating current runs through the air-coil embedded in a housing. The electromagnetic coil section induces eddy currents in the conductive object whose position is to be measured, thus absorbing energy from the oscillating circuit. Depending on the clear‐ ance, the inductance of the coil varies, and external electronic circuitry converts this varia‐ tion into an output signal. The usual modulation frequencies lie in a range of 1 - 2 MHz, resulting in useful measuring frequency ranges of 0 Hz up to approximately 20 kHz.
