**5. Simulation results**

In order to verify the dynamic behavior of the rotor speed controller, active unbalance control scheme and estimation of the unbalance forces, some numerical simulations were carried out using the numerical parameters shown in Table 1.

The performance of the rotor speed controller (18) was evaluated for the tracking of the smooth speed reference profile *<sup>ω</sup>*∗(*t*) shown in Fig. 3, which allows to take the rotor from


**Table 1.** Rotor System Parameters.

an initial speed *<sup>ω</sup>*¯ <sup>1</sup> for *<sup>t</sup>* ≤ *<sup>T</sup>*<sup>1</sup> to the desired final operation speed *<sup>ω</sup>*¯ <sup>2</sup> for *<sup>t</sup>* ≥ *<sup>T</sup>*2. In general, the unbalance response has more interest when the rotor is running above its first critical speeds *<sup>ω</sup>cr*1*<sup>x</sup>* <sup>=</sup> <sup>√</sup>*kx*/*<sup>m</sup>* <sup>=</sup> 223.76 rad/s <sup>=</sup> 2136.8 rpm and *<sup>ω</sup>cr*1*<sup>y</sup>* <sup>=</sup> �*ky*/*<sup>m</sup>* <sup>=</sup> 230.79 rad/s <sup>=</sup> 2203.9 rpm.

The speed profile specified for the rotor system is described by

$$\omega^\*(t) = \begin{cases} \bar{\omega}\_1 \text{ for } 0 \le t < T\_1 \\ \bar{\omega}\_1 + (\bar{\omega}\_2 - \bar{\omega}\_1) \,\psi(t, T\_1, T\_2) \text{ for } T\_1 \le t \le T\_2 \\ \bar{\omega}\_2 \text{ for } t > T\_2 \end{cases} \tag{29}$$

In Fig. 4 the robust and efficient performance of the PI speed controller (18) is shown. Here, the active unbalance control scheme (14) is not performed, i.e., *ux* = *uy* ≡ 0. Therefore, some irregularities of the control torque action can be observed when the rotor passes through its first critical speeds. Fortunately, the presented speed controller results quite robust against

It is important to note that the control gains were selected to get a closed-loop rotor speed

<sup>2</sup> + 2*ζrωnrs* + *ω*<sup>2</sup>

0 2 4 6 8 10 12

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t[s]

Fig. 5 depicts the open-loop rotor unbalance response while the rotor is taken from the rest initial speed (*ω*¯ <sup>1</sup> = 0 rad/s) to the operating speed (*ω*¯ 2= 300 rad/s) above its first critical speeds by using the PI rotor speed controller (18). The presence of high vibration amplitude levels (above 9 mm) at the resonant peaks can be observed. Note in Fig. 6 that the centrifugal forces induced by the rotor unbalance are quite significant. Thus, the active rotor balancing

On the other hand, Figs. 7-9 depict the closed-loop rotor-bearing system response by using simultaneously the disturbance observer-based active unbalance control scheme (14), PI rotor speed controller (18) and disturbance observer (25). One can see in Fig. 7 the robust performance of the PI speed controller (18), achieving an effective tracking of the smooth speed reference profile (29). Since the rotor unbalance-induced torque perturbation input signal *ξw* is canceled by the active balancing controllers (14), a smooth curve of the control torque is accomplished, eliminating the irregularities presented in the control torque

controllers (14) should be actively compensate those perturbation forces in real time.

*nr*

Estimation and Active Damping of Unbalance Forces in Jeffcott-Like Rotor-Bearing Systems

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43

the bounded torque perturbation input signal *ξw* induced by the rotor unbalance.

*P<sup>ω</sup>* (*s*) = *s*

**Figure 4.** Rotor system response using local PI speed controller without active unbalance control.

dynamics having a Hurwitz characteristic polynomial:

with *ωnr* = 15 rad/s and *ζ<sup>r</sup>* = 0.7071.

−0.4 −0.2 0 0.2 0.4

> 0 0.5 1 1.5

response without active unbalance control (4).

τ [Nm]

ω

ξw [Nm]

[rad/s]

where *ω*¯ <sup>1</sup> = 0 rad/s, *ω*¯ <sup>2</sup> = 300 rad/s = 2864.8 rpm, *T*<sup>1</sup> = 0 s, *T*<sup>2</sup> = 10 s and *ψ* (*t*, *T*1, *T*2) is a Bézier polynomial defined as

$$\psi(t) = \left(\frac{t - T\_1}{T\_2 - T\_1}\right)^5 \left[r\_1 - r\_2 \left(\frac{t - T\_1}{T\_2 - T\_1}\right) + r\_3 \left(\frac{t - T\_1}{T\_2 - T\_1}\right)^2 - \dots - r\_6 \left(\frac{t - T\_1}{T\_2 - T\_1}\right)^5\right]$$

with constants *r*<sup>1</sup> = 252, *r*<sup>2</sup> = 1050, *r*<sup>3</sup> = 1800, *r*<sup>4</sup> = 1575, *r*<sup>5</sup> = 700 and *r*<sup>6</sup> = 126.

**Figure 3.** Smooth reference profiles for the rotor speed and acceleration, *ω*∗(*t*) and *ω*˙ <sup>∗</sup> (*t*).

In Fig. 4 the robust and efficient performance of the PI speed controller (18) is shown. Here, the active unbalance control scheme (14) is not performed, i.e., *ux* = *uy* ≡ 0. Therefore, some irregularities of the control torque action can be observed when the rotor passes through its first critical speeds. Fortunately, the presented speed controller results quite robust against the bounded torque perturbation input signal *ξw* induced by the rotor unbalance.

It is important to note that the control gains were selected to get a closed-loop rotor speed dynamics having a Hurwitz characteristic polynomial:

$$P\_{\omega} \left( \mathbf{s} \right) = \mathbf{s}^2 + 2\mathsf{Z}\_{\mathsf{F}}\omega\_{nr}\mathbf{s} + \omega\_{nr}^2$$

with *ωnr* = 15 rad/s and *ζ<sup>r</sup>* = 0.7071.

14 Vibration Control

2203.9 rpm.

**Table 1.** Rotor System Parameters.

Bézier polynomial defined as

� *<sup>t</sup>* − *<sup>T</sup>*<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>−</sup> *<sup>T</sup>* <sup>1</sup>

*ψ* (*t*) =

˙ω∗ (t) [rad/s2]

ω∗(t) [rad/s] *m* = 3.85 kg *cy* = 14 N s/m *u* = 222 *µ*m *kx* = 1.9276 × 105 N/m *d* = 0.020 m *β* = *<sup>π</sup>*

*<sup>ω</sup>*¯ <sup>1</sup> for 0 ≤ *<sup>t</sup>* < *<sup>T</sup>*<sup>1</sup>

� *<sup>t</sup>* − *<sup>T</sup>*<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>−</sup> *<sup>T</sup>* <sup>1</sup>

with constants *r*<sup>1</sup> = 252, *r*<sup>2</sup> = 1050, *r*<sup>3</sup> = 1800, *r*<sup>4</sup> = 1575, *r*<sup>5</sup> = 700 and *r*<sup>6</sup> = 126.

*ω*¯ <sup>2</sup> for *t* > *T*<sup>2</sup>

*ky* = 2.0507 × 105 N/m *l* = 0.7293 m

The speed profile specified for the rotor system is described by

 

*<sup>ω</sup>*∗(*t*) =

�<sup>5</sup> �

*<sup>r</sup>*<sup>1</sup> − *<sup>r</sup>*<sup>2</sup>

**Figure 3.** Smooth reference profiles for the rotor speed and acceleration, *ω*∗(*t*) and *ω*˙ <sup>∗</sup> (*t*).

*cx* <sup>=</sup> 12 N s/m *<sup>r</sup>*disk <sup>=</sup> 0.076 m *<sup>c</sup><sup>ϕ</sup>* <sup>=</sup> 1.5 <sup>×</sup>10−<sup>3</sup> Nm s/rad

an initial speed *<sup>ω</sup>*¯ <sup>1</sup> for *<sup>t</sup>* ≤ *<sup>T</sup>*<sup>1</sup> to the desired final operation speed *<sup>ω</sup>*¯ <sup>2</sup> for *<sup>t</sup>* ≥ *<sup>T</sup>*2. In general, the unbalance response has more interest when the rotor is running above its first critical speeds *<sup>ω</sup>cr*1*<sup>x</sup>* <sup>=</sup> <sup>√</sup>*kx*/*<sup>m</sup>* <sup>=</sup> 223.76 rad/s <sup>=</sup> 2136.8 rpm and *<sup>ω</sup>cr*1*<sup>y</sup>* <sup>=</sup> �*ky*/*<sup>m</sup>* <sup>=</sup> 230.79 rad/s <sup>=</sup>

where *ω*¯ <sup>1</sup> = 0 rad/s, *ω*¯ <sup>2</sup> = 300 rad/s = 2864.8 rpm, *T*<sup>1</sup> = 0 s, *T*<sup>2</sup> = 10 s and *ψ* (*t*, *T*1, *T*2) is a

� +*r*<sup>3</sup>

*<sup>ω</sup>*¯ <sup>1</sup> + (*ω*¯ <sup>2</sup> − *<sup>ω</sup>*¯ <sup>1</sup>) *<sup>ψ</sup>* (*t*, *<sup>T</sup>*1, *<sup>T</sup>*2) for *<sup>T</sup>*<sup>1</sup> ≤ *<sup>t</sup>* ≤ *<sup>T</sup>*<sup>2</sup>

� *<sup>t</sup>* − *<sup>T</sup>*<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>−</sup> *<sup>T</sup>* <sup>1</sup>

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

�2

− ... − *<sup>r</sup>*<sup>6</sup>

� *<sup>t</sup>* − *<sup>T</sup>*<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>−</sup> *<sup>T</sup>* <sup>1</sup>

�5 �

<sup>4</sup> rad

(29)

**Figure 4.** Rotor system response using local PI speed controller without active unbalance control.

Fig. 5 depicts the open-loop rotor unbalance response while the rotor is taken from the rest initial speed (*ω*¯ <sup>1</sup> = 0 rad/s) to the operating speed (*ω*¯ 2= 300 rad/s) above its first critical speeds by using the PI rotor speed controller (18). The presence of high vibration amplitude levels (above 9 mm) at the resonant peaks can be observed. Note in Fig. 6 that the centrifugal forces induced by the rotor unbalance are quite significant. Thus, the active rotor balancing controllers (14) should be actively compensate those perturbation forces in real time.

On the other hand, Figs. 7-9 depict the closed-loop rotor-bearing system response by using simultaneously the disturbance observer-based active unbalance control scheme (14), PI rotor speed controller (18) and disturbance observer (25). One can see in Fig. 7 the robust performance of the PI speed controller (18), achieving an effective tracking of the smooth speed reference profile (29). Since the rotor unbalance-induced torque perturbation input signal *ξw* is canceled by the active balancing controllers (14), a smooth curve of the control torque is accomplished, eliminating the irregularities presented in the control torque response without active unbalance control (4).

with *ωnx* = *ωny* = 10 rad/s and *ζ<sup>x</sup>* = *ζ<sup>x</sup>* = 0.7071.

compensation of the estimated unbalance force signals *ξ*

than the rotordynamics and, therefore, are specified as

*s* + *po*,*<sup>i</sup>*

**Figure 7.** Rotor system response using local PI speed controller with active unbalance control.

**Figure 8.** Closed-loop rotor unbalance response with local PI rotor speed controller.

with desired parameters *po*,*<sup>i</sup>* = *ωo*,*<sup>i</sup>* = 1200 rad/s and *ζo*,*<sup>i</sup>* = 100.

 *s*

*Po*,*<sup>i</sup>* (*s*) =

−0.4 −0.2 0 0.2 0.4

> 0 0.5 1 1.5

−0.01 −0.005 0 0.005 0.01

−0.01 −0.005 0 0.005 0.01

> 0.005 0.01

yu [m]

0

x [m]

y [m]

τ [Nm]

ω

ξw [Nm]

[rad/s]

Fig. (9) describes the active vibration control scheme response, which applies the active

The characteristic polynomials, assigned to the observation error dynamics, must be faster

<sup>2</sup> + 2*ζo*,*iωo*,*is* + *ω*<sup>2</sup>

0 2 4 6 8 10 12

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

0 2 4 6 8 10 12

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

*<sup>x</sup>* and *<sup>ξ</sup>*

Estimation and Active Damping of Unbalance Forces in Jeffcott-Like Rotor-Bearing Systems

*o*,*i* 3

, *i* = *x*, *y*.

*<sup>y</sup>* shown in Fig. (10).

http://dx.doi.org/10.5772/51180

45

**Figure 5.** Open-loop rotor unbalance response with local PI rotor speed controller.

**Figure 6.** Rotor unbalance forces without active unbalance control.

The closed-loop rotor unbalance response is described in Fig. 8. The active unbalance suppression can be clearly noted. In this case, the gains of the active unbalance controllers were selected to get a closed-loop system dynamics having the Hurwitz characteristic polynomials:

$$\begin{aligned} P\_{\mathfrak{X}}\left(s\right) &= s^2 + 2\zeta\_{\mathfrak{X}}\omega\_{\mathfrak{n}\mathfrak{x}}s + \omega\_{\mathfrak{n}\mathfrak{x}}^2, \\ P\_{\mathfrak{Y}}\left(s\right) &= s^2 + 2\zeta\_{\mathfrak{Y}}\omega\_{\mathfrak{n}\mathfrak{y}}s + \omega\_{\mathfrak{n}\mathfrak{y}}^2. \end{aligned}$$

with *ωnx* = *ωny* = 10 rad/s and *ζ<sup>x</sup>* = *ζ<sup>x</sup>* = 0.7071.

16 Vibration Control

x [m]

y [m]

−0.01 −0.005 0 0.005 0.01

−0.01 −0.005 0 0.005 0.01

0.01

0.005

−100 −50 0 50 100

−100 −50 0 50 100

**Figure 6.** Rotor unbalance forces without active unbalance control.

ξ<sup>x</sup> [N]

ξ<sup>y</sup> [N]

polynomials:

yu [m]

0

**Figure 5.** Open-loop rotor unbalance response with local PI rotor speed controller.

0 2 4 6 8 10 12

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

The closed-loop rotor unbalance response is described in Fig. 8. The active unbalance suppression can be clearly noted. In this case, the gains of the active unbalance controllers were selected to get a closed-loop system dynamics having the Hurwitz characteristic

<sup>2</sup> + 2*ζxωnxs* + *ω*<sup>2</sup>

<sup>2</sup> + 2*ζyωnys* + *ω*<sup>2</sup>

*nx*

*ny*

*Px* (*s*) = *s*

*Py* (*s*) = *s*

Fig. (9) describes the active vibration control scheme response, which applies the active compensation of the estimated unbalance force signals *ξ <sup>x</sup>* and *<sup>ξ</sup> <sup>y</sup>* shown in Fig. (10).

The characteristic polynomials, assigned to the observation error dynamics, must be faster than the rotordynamics and, therefore, are specified as

$$P\_{o,i}\left(\mathbf{s}\right) = \left(\mathbf{s} + p\_{o,i}\right)\left(\mathbf{s}^2 + 2\mathsf{Z}\_{o,i}\omega\_{o,i}\mathbf{s} + \omega\_{o,i}^2\right)^3, \ \mathbf{i} = \mathbf{x}, \mathbf{y}.$$

with desired parameters *po*,*<sup>i</sup>* = *ωo*,*<sup>i</sup>* = 1200 rad/s and *ζo*,*<sup>i</sup>* = 100.

**Figure 7.** Rotor system response using local PI speed controller with active unbalance control.

**Figure 8.** Closed-loop rotor unbalance response with local PI rotor speed controller.

perturbation force signals and velocities of the rotor center coordinates based on Luenberger linear state observers has also been proposed. In the state observer design process, the perturbation force signals were locally approximated by a family of Taylor time-polynomials of fourth degree. Therefore, each perturbation signal was locally described by a state space-based linear mathematical model of fifth order. Then, an extended lineal mathematical model was obtained to locally describe the dynamics of the perturbed rotor system to be used in the design of the disturbance and state observer. In addition, a PI rotor speed controller was proposed to perform robust tracking tasks of smooth rotor speed reference profiles described by Bézier interpolation polynomials. Simulations results show the robust and efficient performance of the active vibration control scheme and rotor speed controller proposed in this chapter, as well as the fast and effective estimation of the perturbation force signals, when the rotor system is taken from a rest initial speed to an operation speed above its first critical velocities. The proposed methodology can be applied for more complex and realistic rotor-bearing systems (e.g., more disks, turbines, shaft geometries), finite element models, monitoring and fault diagnosis quite common in industrial rotating machinery.

Estimation and Active Damping of Unbalance Forces in Jeffcott-Like Rotor-Bearing Systems

http://dx.doi.org/10.5772/51180

47

1 Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Departamento de Energia,

2 Centro de Investigacion y de Estudios Avanzados del I.P.N., Departamento de Ingenieria

3 Universidad Tecnologica de la Mixteca, Instituto de Electronica y Mecatronica, Huajuapan

[1] Arias-Montiel, M. & Silva-Navarro, G. (2010). Active Unbalance Control in a Two Disks Rotor System Using Lateral Force Actuators, *Proceeding of 7th International Conference on Electrical Engineering, Computing Science and Automatic Control*, pp. 440-445, Tuxtla

[2] Arias-Montiel, M. & Silva-Navarro, G. (2010b). Finite Element Modelling and Unbalance Compensation for an Asymmetrical Rotor-Bearing System with Two Disks, *In: New Trends in Electrical Engineering, Automatic Control, Computing and Communication Sciences, Edited by C.A. Coello-Coello, Alex Pozniak, José A. Moreno-Cadenas and Vadim*

[3] Cabrera-Amado, M. & Silva-Navarro, G. (2010). Semiactive Control for the Unbalance Compensation in a Rotor-Bearing System, *In: New Trends in Electrical Engineering, Automatic Control, Computing and Communication Sciences, Edited by C.A. Coello-Coello, Alex Pozniak, José A. Moreno-Cadenas and Vadim Azhmyakov*, pp. 143-158, Logos Verlag

*Azhmyakov*, pp. 127-141, Logos Verlag Berlin GmbH, Germany.

**Author details**

Mexico, D.F., Mexico

de Leon, Oaxaca, Mexico

Gutierrez, México.

Berlin GmbH, Germany.

**References**

Francisco Beltran-Carbajal1,

Gerardo Silva-Navarro2 and Manuel Arias-Montiel<sup>3</sup>

Electrica, Seccion de Mecatronica, Mexico, D.F., Mexico

**Figure 9.** Response of active unbalance Controllers.

**Figure 10.** Estimates of the closed-loop rotor unbalance force signals using disturbance observer (25).

## **6. Conclusions**

In this chapter, we have proposed a PD-like active vibration control scheme for robust and efficient suppression of unbalance-induced synchronous vibrations in variable-speed Jeffcott-like non-isotropic rotor-bearing systems of three degrees of freedom using only measurements of the radial displacement close to the disk. In this study, we have considered the application of an active suspension device, which is based on two linear electromechanical actuators and helicoidal compression springs, to provide the control forces required for on-line balance of the rotor system. The presented control approach is mainly based on the compensation of bounded perturbation force signals induced by the rotor unbalance, and the specification of the desired closed-loop rotor-bearing system dynamics (viscous damping ratios and natural frequencies). A robust and fast estimation scheme of the perturbation force signals and velocities of the rotor center coordinates based on Luenberger linear state observers has also been proposed. In the state observer design process, the perturbation force signals were locally approximated by a family of Taylor time-polynomials of fourth degree. Therefore, each perturbation signal was locally described by a state space-based linear mathematical model of fifth order. Then, an extended lineal mathematical model was obtained to locally describe the dynamics of the perturbed rotor system to be used in the design of the disturbance and state observer. In addition, a PI rotor speed controller was proposed to perform robust tracking tasks of smooth rotor speed reference profiles described by Bézier interpolation polynomials. Simulations results show the robust and efficient performance of the active vibration control scheme and rotor speed controller proposed in this chapter, as well as the fast and effective estimation of the perturbation force signals, when the rotor system is taken from a rest initial speed to an operation speed above its first critical velocities. The proposed methodology can be applied for more complex and realistic rotor-bearing systems (e.g., more disks, turbines, shaft geometries), finite element models, monitoring and fault diagnosis quite common in industrial rotating machinery.

## **Author details**

18 Vibration Control

−100 −50 0 50 100

−100 −50 0 50 100

−100 −50 0 50 100

−100 −50 0 50 100

**6. Conclusions**

ξ<sup>x</sup> [N]

ξ<sup>y</sup> [N]

**Figure 9.** Response of active unbalance Controllers.

uy [N]

ux [N]

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

0 2 4 6 8 10 12

0 2 4 6 8 10 12

t [s]

In this chapter, we have proposed a PD-like active vibration control scheme for robust and efficient suppression of unbalance-induced synchronous vibrations in variable-speed Jeffcott-like non-isotropic rotor-bearing systems of three degrees of freedom using only measurements of the radial displacement close to the disk. In this study, we have considered the application of an active suspension device, which is based on two linear electromechanical actuators and helicoidal compression springs, to provide the control forces required for on-line balance of the rotor system. The presented control approach is mainly based on the compensation of bounded perturbation force signals induced by the rotor unbalance, and the specification of the desired closed-loop rotor-bearing system dynamics (viscous damping ratios and natural frequencies). A robust and fast estimation scheme of the

**Figure 10.** Estimates of the closed-loop rotor unbalance force signals using disturbance observer (25).

Francisco Beltran-Carbajal1, Gerardo Silva-Navarro2 and Manuel Arias-Montiel<sup>3</sup>

1 Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Departamento de Energia, Mexico, D.F., Mexico

2 Centro de Investigacion y de Estudios Avanzados del I.P.N., Departamento de Ingenieria Electrica, Seccion de Mecatronica, Mexico, D.F., Mexico

3 Universidad Tecnologica de la Mixteca, Instituto de Electronica y Mecatronica, Huajuapan de Leon, Oaxaca, Mexico

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**Chapter 3**

**Rotordynamic Stabilization of Rotors on**

In the last decades the advance in the semiconductors technology for power electronics has dictated a growing interest for high rotational speed machines. The use of high rotational speeds allows increasing the power density of the machine, but introduces some critical as‐ pects from the mechanical point of view. One of the most critical issues to be dealt with is the difficulty in operating common mechanical bearings in this condition. For this reason al‐ ternatives for classical ball and roller bearings must be found. In this context, active magnet‐ ic bearings represent an advantageous alternative because they are capable of supporting the rotating shaft in absence of contact. Nevertheless, the high cost associated with this kind

A promising system for supporting high rotational speed machines in absence of contact and with relatively low costs, widening the range of applications, is the electrodynamic suspension of rotors [1], [2], [3], [4], [5]. Systems capable of realizing this concept are commonly referred to as electrodynamic bearings (EDB). They exploit repulsive forces due to eddy currents arising between conductors in motion relative to a magnetic field. The supporting forces are generat‐ ed in a completely passive process, thus representing an increase in the overall reliability of the suspension with respect to active magnetic bearings. Nevertheless, electrodynamic bearings have drawbacks. The eddy current forces that provide levitation produce an energy dissipa‐

Because the rotor may present an unstable behavior, it is necessary to study the dynamic re‐ sponse of the suspension in order to guarantee stable operation in the working range of speed. This can be achieved by introducing nonrotating damping in the system, but the choice of the damping elements is not obvious, requiring an accurate modeling phase. The present paper presents the development of a dynamic model of the entire suspension that is

> © 2012 Detoni 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,

© 2012 Detoni 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.

distribution, and reproduction in any medium, provided the original work is properly cited.

tion that may cause negative damping resulting in rotordynamic instability.

J. G. Detoni, F. Impinna, N. Amati and A. Tonoli

Additional information is available at the end of the chapter

**Electrodynamic Bearings**

http://dx.doi.org/10.5772/51350

of system reduces their applicability.

**1. Introduction**
