**3. Modeling**

The dynamic behavior of the rotor can be described by means of different models. This sec‐ tion describes the modeling techniques and the adopted assumptions, how acting forces and displacements are ordered and selected, the equation of motion used for the dynamic de‐ scription and the variable used to describe the models.

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‐ 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.

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‐ ning rotation) should be small.

## **3.1. Model Block Diagram**

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

In addition, depending on the application, speeds, currents, flux densities and temperatures

When selecting the displacement sensors, depending on the application of the magnetic bearing, measuring range, linearity, sensitivity, resolution, and frequency range are to be

**•** Noise immunity against other sensors, magnetic alternating fields of the electromagnets,

**•** Temperature range, temperature drift of the zero point and sensitivity;

**•** Environmental factors such as dust, aggressive media, vacuum, or radiation;

**•** Electrical factors such as grounding issues associated with capacitive sensors.

**•** Eddy Current Radial Displacement Sensor on a PCB (Transverse Flux Sensor)

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,

The dynamic behavior of the rotor can be described by means of different models. This sec‐ tion describes the modeling techniques and the adopted assumptions, how acting forces and displacements are ordered and selected, the equation of motion used for the dynamic de‐

resulting in useful measuring frequency ranges of 0 Hz up to approximately 20 kHz.

electromagnetic disturbances from switched amplifiers;

The most important displacement sensors technologies are:

scription and the variable used to describe the models.

**•** Mechanical factors such as shock and vibration;

frequency or with multiples thereof.

10 Advances in Vibration Engineering and Structural Dynamics

taken into account as well as:

**•** Inductive sensors;

**•** Eddy-current sensors;

**•** Capacitive sensors;

**•** Magnetic sensors.

**3. Modeling**

are to be measured in magnetic bearing systems.

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‐ motive force in the motor or magnetic bearings.

## *3.1.1. Model inputs / outputs*

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 motor, used to generate rotation torque (not included in this kind of model where the spin speed is assumed to be constant), can also act two forces, in radial direction, while is not ca‐ pable to generate a force in the Z direction. According to this, vector fMOT has two compo‐ nents acting in the center of the motor.

**AMB Motor External**

*F F* ì ü <sup>=</sup> í ý î þ

MY

Referring to Figure 8, a set of outputs is used for measurements (the spin speed *Ω* and the dis‐ placements on the sensor qSENS) and another set is used for intrinsical feedback (displace‐

**Sensors AMB Motor**

*x y*

AMB AMB2

**Input Output X-Y Behaviour Z Behaviour X-Y Behaviour Z Behaviour**

*z*

x

*y*

AMB1 AMB1

ì ü ï ï

> AMB2 AMB3

ï ï î þ

*<sup>x</sup> <sup>z</sup> <sup>y</sup>*

ï ï ï ïï ï ì ü = = í ýí ý

ï ï ï ï î þ ï ï

ments and velocities on AMBs qAMB and *q*˙ AMB, and on the motor qMOT and *q*˙ MOT).

EXTx

ï ï ì ü ï ï = = í ýí ý ï ï ï ï î þ î þ

Rotors on Active Magnetic Bearings: Modeling and Control Techniques

ì ü

*F F*

*F*

EXTz

EXT EXTy

AMBxy

**q**

**q**

1 AMB1 MOT AMB2 SENS2

*SENS*

ì ü ï ï ï ï ï ï ï ï ï ï ï ï ï ï <sup>=</sup> í ý ï ï ï ï ï ï ï ï ï ï ï ï ï ï î þ

*x x x x x y y y y y*

SENS1 AMB1 MOT AMB2 SENS2

**y** SENS3

*z*

**y**

AMB3

**f**

EXTxy

(1)

13

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EXTz

MOT

(2)

(3)

*x y z*

ì ü ï ï <sup>=</sup> í ý

MOT

ï ï î þ

MOT MOT

ΑΜΒ3

*z z* ì ü <sup>=</sup> í ý î þ

*F*

**f**

A1X A1Y

ì ü ï ï

*F F F*

> A2Y A3Z

ï ï î þ

SENS1 SENS1

ì ü ï ï

*x y*

> SENS2 SENS3

ï ï î þ

*<sup>x</sup> <sup>z</sup> <sup>y</sup>*

ï ï ï ïï ï ì ü = = í ýí ý

ï ï ï ï î þ ï ï

SENS SENS2

**q**

**Table 4.** Rotor outputs.

xy

*z*

A1X EXTx MX A2X

ì ü ï ï ï ï ï ï ï ï ï ï <sup>=</sup> í ý ï ï ï ï ï ï ï ï ï ï î þ

*F F F F F F F F*

A1Y EXTx MY A2Y

**Table 5.** Inputs / Outputs vector orders.

**f** A3Z

z

**f**

EXTz

*F F* ì ü <sup>=</sup> í ý î þ

*F F*

AMB A2X

**Table 3.** Rotor Inputs.

AMBxy

MOT

**f**

Table 4 reports displacements and velocities (outputs) on the rotor.

SENSxy

**q**

**q**

SENS3

A3Z

*F*

**f f** MX

ï ï ï ïï ï ì ü = = í ýí ý

ï ï ï ï î þ ï ï

**Figure 8.** Rotor model block diagram.

In order to simplify the description of the system the generic external force are supposed to act directly to the center of mass of the rotor; these three forces are oriented as the plant ref‐ erence frame. These components are the resultant of any external force, such as impact forces. While the X-Y behaviour is uncoupled from the Z behavior, acting forces due to AMBs and external can be rewritten dividing the forces acting in X-Y plane from forces act‐ ing on Z axis. Table 3 reports acting forces (inputs) on the rotor.

**Figure 9.** Actuation forces and sensors position.

**Table 3.** Rotor Inputs.

motor, used to generate rotation torque (not included in this kind of model where the spin speed is assumed to be constant), can also act two forces, in radial direction, while is not ca‐ pable to generate a force in the Z direction. According to this, vector fMOT has two compo‐

In order to simplify the description of the system the generic external force are supposed to act directly to the center of mass of the rotor; these three forces are oriented as the plant ref‐ erence frame. These components are the resultant of any external force, such as impact forces. While the X-Y behaviour is uncoupled from the Z behavior, acting forces due to AMBs and external can be rewritten dividing the forces acting in X-Y plane from forces act‐

ing on Z axis. Table 3 reports acting forces (inputs) on the rotor.

nents acting in the center of the motor.

12 Advances in Vibration Engineering and Structural Dynamics

**Figure 8.** Rotor model block diagram.

**Figure 9.** Actuation forces and sensors position.

Referring to Figure 8, a set of outputs is used for measurements (the spin speed *Ω* and the dis‐ placements on the sensor qSENS) and another set is used for intrinsical feedback (displace‐ ments and velocities on AMBs qAMB and *q*˙ AMB, and on the motor qMOT and *q*˙ MOT).

Table 4 reports displacements and velocities (outputs) on the rotor.

**Table 4.** Rotor outputs.


**Table 5.** Inputs / Outputs vector orders.

Input/Output vector reported in Table 5. and ordered to be compliant with blocks that gen‐ erate such forces. In order to address the modeling technique (especially FE based), input/ output vector should be reordered in the way reported in Table 5.

The dynamic behavior of the rotor can be described by mean of the equations of motion (EoM). In the following section the equations used for the model is described. The typical equations are reported for a generic rotor model and then applied to a rigid analytical model and to the FE model.While the spin speed is constant the X-Y behavior is uncoupled from Z behavior in the same way the equations can be separated.

*X-Y Behavior*

Equation of Motion:

$$\begin{aligned} \mathbf{M}\_{\text{xy}} \ddot{\mathbf{q}}\_{\text{xy}}(t) + \left( \mathbf{L}\_{\text{xy}} + \Omega \mathbf{G}\_{\text{xy}} \right) \dot{\mathbf{q}}\_{\text{xy}}(t) + \left( \mathbf{K}\_{\Omega \text{cup}} + \Omega^2 \mathbf{K}\_{\Omega \text{2},\text{xy}} + \Omega \mathbf{H}\_{\text{xy}} \right) \mathbf{q}\_{\text{xy}}(t) &= \\ = \mathbf{f}\_{\text{xy}} + \Omega^2 \mathbf{f}\_{\text{umb}} \begin{bmatrix} \sin(\Omega t) \\ \cos(\Omega t) \end{bmatrix} + \mathbf{S}\_{\text{xy}} \mathbf{f}\_{\text{xy}}(t) \end{aligned} \tag{1}$$

Measure Equation:

$$\begin{Bmatrix} \mathbf{y}\_{\text{xy}}(t) \\ \dot{\mathbf{y}}\_{\text{xy}}(t) \end{Bmatrix} = \mathbf{S}\_{\text{avg}} \begin{Bmatrix} \mathbf{q}\_{\text{xy}}(t) \\ \dot{\mathbf{q}}\_{\text{xy}}(t) \end{Bmatrix} \tag{2}$$

**Name Description**

*f <sup>s</sup>* Static forces *R* Rotation Matrix *f umb* Unbalance forces *f* (*t*) External forces

**Table 6.** Matrices names and description.

tem as reported in Figure 10.

**3.2. 4dof model**

*S<sup>i</sup>* Input selection matrix *y*(*t*) Output displacements *S<sup>o</sup>* Output selection matrix

**Figure 10.** d.o.f. model with generalized displacements and forces.

*M* Mass (symmetric) matrix *L* Damping (symmetric) matrix

*G* Gyroscopic (skew-symmetric) matrix

*H* Circulatory (skew-symmetric) matrix

*K*Ω<sup>0</sup> Stiffness (symmetric) matrix: spin speed independent *K*Ω<sup>2</sup> Stiffness (symmetric) matrix: spin speed dependent

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15

Φ Selected eigenvector for modal (MK) reduction

Generic Eom for rotors previously described, can be applied to a rigid analytical model based on the 4 d.o.f theory (for X-Y behavior), with an additional d.o.f. for the Z behav‐ ior. In this model the equation of motion are develop in a center of mass coordinate sys‐

*Z Behavior*

Equation of Motion:

$$\mathbf{M}\_z \ddot{\mathbf{q}}\_z(t) + \mathbf{L}\_z \dot{\mathbf{q}}\_z(t) + \left(\mathbf{K}\_{\Omega 0z} + \Omega^2 \mathbf{K}\_{\Omega 2z}\right) \mathbf{q}\_z(t) = \mathbf{f}\_{sz} + \mathbf{S}\_{\mu} \mathbf{f}\_z(t) \tag{3}$$

Measure Equation:

$$\begin{Bmatrix} \mathbf{y}\_{\varepsilon}(t) \\ \dot{\mathbf{y}}\_{\varepsilon}(t) \end{Bmatrix} = \mathbf{S}\_{\alpha \varepsilon} \begin{Bmatrix} \mathbf{q}\_{\varepsilon}(t) \\ \dot{\mathbf{q}}\_{\varepsilon}(t) \end{Bmatrix} \tag{4}$$

Refer to Table 6 for matrices naming and description.



**Table 6.** Matrices names and description.

### **3.2. 4dof model**

Input/Output vector reported in Table 5. and ordered to be compliant with blocks that gen‐ erate such forces. In order to address the modeling technique (especially FE based), input/

The dynamic behavior of the rotor can be described by mean of the equations of motion (EoM). In the following section the equations used for the model is described. The typical equations are reported for a generic rotor model and then applied to a rigid analytical model and to the FE model.While the spin speed is constant the X-Y behavior is uncoupled from Z

( ) ( ) <sup>2</sup>

*(t) <sup>t</sup>*

*xy xy oxy xy xy*

( ) <sup>2</sup>

*z z oz z z*

*t (t) t (t)* ì ü ìü í ý íý = î þ îþ **y q <sup>S</sup>**

0 2 ( *zz zz <sup>z</sup> z z sz iz z (t) (t) (t) t)* **Mq Lq K K q f S f** + + +W = + W W && & (3)

*t (t) t (t)* ì ü ìü ï ï ïï í ý íý = ï ï ïï î þ îþ **y q <sup>S</sup>**

*(t) (t) (t)*

+ +W + +W +W = W W

*xy xy xy xy xy xy xy xy xy*

**Mq L G q K K H q**

0 2

**y q** & & (2)

**y q** & & (4)

(1)

output vector should be reordered in the way reported in Table 5.

behavior in the same way the equations can be separated.

<sup>2</sup> sin( ) cos( )

**f f S f**

= +W í ý +

&& &

14 Advances in Vibration Engineering and Structural Dynamics

*sxy umb ixy xy*

*t*

( ) ( )

> ( ) ( )

Refer to Table 6 for matrices naming and description.

*q*(*t*) Generalized displacements

**Name Description** Ω Spin Speed

ì ü W

î þ W

*X-Y Behavior*

Equation of Motion:

Measure Equation:

Equation of Motion:

Measure Equation:

*Z Behavior*

Generic Eom for rotors previously described, can be applied to a rigid analytical model based on the 4 d.o.f theory (for X-Y behavior), with an additional d.o.f. for the Z behav‐ ior. In this model the equation of motion are develop in a center of mass coordinate sys‐ tem as reported in Figure 10.

**Figure 10.** d.o.f. model with generalized displacements and forces.


The physical properties used in the model are:

**Table 7.** Physical properties of rigid analytical model.

#### *X-Y Behavior*

The generalized displacements vector is composed by the center of mass coordinates:

$$\mathbf{q}\_{\rm xy}(t) = \begin{bmatrix} \mathbf{x}\_G \\ \mathbf{y}\_G \\ \mathbf{q}\_{\mathbf{x}\_G} \\ \mathbf{q}\_{\rm y} \end{bmatrix} \tag{5}$$

The measure equation has the same structure reported in (5).

The measure equation has the same structure reported in (7).

0 00 0 0

<sup>=</sup> - -

'

*b bb*

'

= = ê ú ê ú ë û

**Table 8.** Input/Output selection matrices for rigid analytical model.

*oxy oxy*

é ù

**S 0 S S 0 S**

'

**3.3. Flexible Rotor Model (FE)**

reported in equation (4) to (7).

*3.3.1. Full Model*

the following order:

equation (9) and (10) with their relative measure equation should be:

1 11 1 00 0 0 0 00 0 11 1 1

é ù ê ú

ë û -

In order to be compliant to input/output vector described in Table 8 the selection matrices of

1 2

é ù ê ú -

*A MA*

*b bb*

2

10 0 , 01 0

1

*A M A S*

01 0 01 0 01 0 01 0


A flexible model is here described. This model is generated by using a finite element code especially designed for rotating machines (Dynrot). The outputs of this code are the matrices

In the case all the nodes displacements are used the full dynamic behavior is described. The displacements vector *q* collects the translation displacement of the model nodes in

*b b b b b*

2 2

**S** [ ]1 2 1 1 *iz <sup>x</sup>* **S** =

1 2 4 8

0 00 0 0

*A MA x*

<sup>1</sup> 20 8

*oxy S*

*oxy <sup>S</sup> <sup>x</sup>*

**X-Y Behaviour Z Behaviour**

1 1

*S A M A*

*b b b b b*

2

( *mz(t) t)* = + *sz iz z* && **f Sf** (7)

Rotors on Active Magnetic Bearings: Modeling and Control Techniques

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17

4 2

*x*

é ù ê ú = ê ú ê ú ê ú ë û

*oz*

**S**

Under the same assumptions equation (6) becomes:

*Z Behavior*

*ixy*

If the model has no damping the EoM (4) becomes:

$$\begin{aligned} \begin{bmatrix} m & 0 & 0 & 0 \\ 0 & m & 0 & 0 \\ 0 & 0 & J\_{\boldsymbol{\cdot}} & 0 \\ 0 & 0 & 0 & J\_{\boldsymbol{\cdot}} \end{bmatrix} \ddot{\mathbf{q}}\_{\mathcal{U}}(t) + \Omega \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & J\_{\boldsymbol{\rho}} \\ 0 & 0 & -J\_{\boldsymbol{\rho}} & 0 \end{bmatrix} \dot{\mathbf{q}}\_{\mathcal{U}}(t) = \\\ \begin{bmatrix} m \mathbf{g}\_{\boldsymbol{x}} \\ m \mathbf{g}\_{\boldsymbol{y}} \\ 0 \\ 0 \end{bmatrix} + \Omega^{2} \begin{bmatrix} -m\boldsymbol{\omega}\_{\boldsymbol{y}} & m\boldsymbol{\omega}\_{\boldsymbol{x}} \\ m\boldsymbol{\omega}\_{\boldsymbol{x}} & m\boldsymbol{\omega}\_{\boldsymbol{y}} \\ -\mathbf{\mathcal{X}}\left(J\_{\boldsymbol{\cdot}} - J\_{\boldsymbol{\cdot}}\right) & 0 \\ 0 & \mathbf{\mathcal{X}}\left(J\_{\boldsymbol{\cdot}} - J\_{\boldsymbol{\cdot}}\right) \end{bmatrix} \begin{bmatrix} \sin(\Omega t) \\ \cos(\Omega t) \end{bmatrix} + \mathbf{S}\_{\boldsymbol{\text{in}}} \mathbf{f}\_{\boldsymbol{y}}(t) \end{aligned} \tag{6}$$

The measure equation has the same structure reported in (5).

## *Z Behavior*

The physical properties used in the model are:

16 Advances in Vibration Engineering and Structural Dynamics

*m* Mass of the rotor [kg]

**Table 7.** Physical properties of rigid analytical model.

If the model has no damping the EoM (4) becomes:

*gy* Gravity along x and y direction [m/s2]

χ Torque unbalance: angular error [rad] *bA*<sup>1</sup> AMB1 distance from center of mass [m] *bA*<sup>2</sup> AMB2 distance from center of mass [m] *bM* Motor distance from center of mass [m] *bS*<sup>1</sup> Sensor1 distance from center of mass [m] *bS*<sup>2</sup> Sensor2 distance from center of mass [m]

ε*x*, ε*<sup>y</sup>* Static unbalance eccentricity along x and y direction [m]

*Jt* Transversal moment of inertia about any axis in the rotation plane [kgm2] *Jp* Transversal moment of inertia about any axis in the rotation plane [kgm2]

The generalized displacements vector is composed by the center of mass coordinates:

*xy*

( )

*y x <sup>x</sup> x y <sup>y</sup>*

é ù - ì ü ê ú ï ï

e

î þ - ë û

e

*xy xy*

**q q**

&& &

0 0 cos( )

ï ï ì ü W = +W í ý ê ú í ý + ï ï - - î þ <sup>W</sup>

c

*m m mg <sup>t</sup>*

2

0 0

ï ï

000 00 0 0 0 00 00 0 0 00 0 00 0 000 00 0

é ù é ù ê ú ê ú

ê ú ê ú - ë û ë û

*t p t p*

ê ú ê ú + W <sup>=</sup> ê ú ê ú

*mg m m*

*J J*

c

*<sup>m</sup> (t) (t) <sup>J</sup> <sup>J</sup>*

( )

*J J*

 e

 e

*t p*

*ixy xy t p*

*(t) J J <sup>t</sup>*

sin( )

**S f**

(6)

*(t)*

*G G*

**q** (5)

*x y*

j

j

*G G*

ì ü ï ï ï ï <sup>=</sup> í ý ï ï ï ï î þ

*x y*

**Name Description**

*gx*,

*X-Y Behavior*

*m*

Under the same assumptions equation (6) becomes:

$$m\ddot{z}(t) = \mathbf{f}\_{sz} + \mathbf{S}\_{iz}\mathbf{f}\_z(t) \tag{7}$$

The measure equation has the same structure reported in (7).

In order to be compliant to input/output vector described in Table 8 the selection matrices of equation (9) and (10) with their relative measure equation should be:


**Table 8.** Input/Output selection matrices for rigid analytical model.

### **3.3. Flexible Rotor Model (FE)**

A flexible model is here described. This model is generated by using a finite element code especially designed for rotating machines (Dynrot). The outputs of this code are the matrices reported in equation (4) to (7).

#### *3.3.1. Full Model*

In the case all the nodes displacements are used the full dynamic behavior is described. The displacements vector *q* collects the translation displacement of the model nodes in the following order:

## *3.3.1. Reduced Model*

Usually a reduced model is used. A typical reduction method is the modal (MK) reduction where only some modes of vibration are selected.

**3.4. State Space representationofrotordynamicequations**

resentation in the following way, either for X-Y or Z behavior:

() () ()

*ttt*

**0** *I*

<sup>−</sup>1*K*Ω0*xy* <sup>−</sup>*Mxy*

**0 0**

<sup>−</sup>1*Hxy* <sup>−</sup>*Mxy*

**0 0**

<sup>−</sup>1*K*Ω2*xy* **<sup>0</sup>**

*C Soxy Soz D* **0 0**

= +

**y Cx Du**

Where:

*x*(*t*) {

*A*Ω<sup>0</sup>

*A*Ω<sup>1</sup>

*A*Ω<sup>2</sup>

*B*

*q*xy(*t*) *q***˙**xy(*t*)

−*Mxy*

−*Mxy*

−*Mxy*

*f* umb *f* xy

*y***˙**xy(*t*)

**0** *I* **0** *I* **0** *Sixy*

*<sup>u</sup>*(*t*) { *<sup>f</sup>* sxy

*<sup>y</sup>*(*t*) {*y*xy(*t*)

**Table 12.** State Space Representation variables.

**4. Control design and results**

to get a steady and balanced control action.

( ) <sup>2</sup>

**Name X-Y Behaviour Z Behaviour**

} {

<sup>−</sup>1*<sup>L</sup> xy*

} { *<sup>f</sup>* sz

} {*y*z(*t*)

−1 *Gxy*

Dynamic equations (4) to (7) can be reported, with explicit spin speed, in the state space rep‐

*q*z(*t*) *q***˙**z(*t*)

−*M<sup>z</sup>*

−*M<sup>z</sup>*

**0** *I* **0** *Siz*

*f* z }

*y***˙**z(*t*) }

The aim of this section is to explain the steps followed to perform the suspension control design. A conventional decentralized control strategy is illustrated with two nested loops, the inner for current control and the outer for the position.The detailed description of this strategy is followed by the exposition of an off-line electrical centering strategy which is used equalize electrical parameters of the electromagnets on each actuation stage and allows

**0**

& (8)

**0** *I*

<sup>−</sup>1*K*Ω0*<sup>z</sup>* <sup>−</sup>*M<sup>z</sup>*

**0 0**

<sup>−</sup>1*K*Ω2*<sup>z</sup>* **<sup>0</sup>**

} (16)

Rotors on Active Magnetic Bearings: Modeling and Control Techniques

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19

<sup>−</sup>1*<sup>L</sup> <sup>z</sup>*

01 2 ( ) () () ()

*t tt t*

= +W +W WW W +

**x A A A x u Bu**

The displacements to modal transformation are reported in Table 10:

**Table 9.** Generalized coordinates for full FE model.


**Table 10.** Nodal to Modal transformation.

The equations of motion for the reduced model are formerly the same reported in equations (4) to (7), where the nodal displacements is substituted by the modal displacements and ma‐ trices and vector are reported in their modal forms.

Input/output selection matrices should be transformed, starting from FEM matrices, as indi‐ cated in:


**Table 11.** Input / Output matrices conversion from Nodal to Modal.

### **3.4. State Space representationofrotordynamicequations**

Dynamic equations (4) to (7) can be reported, with explicit spin speed, in the state space rep‐ resentation in the following way, either for X-Y or Z behavior:

$$\begin{aligned} \dot{\mathbf{x}}(t) &= \left(\mathbf{A}\_{\Omega 0} + \Omega \mathbf{A}\_{\Omega 1} + \Omega^2 \mathbf{A}\_{\Omega 2}\right) \mathbf{x}(t)\mathbf{u}(t) + \mathbf{B} \mathbf{u}(t) \\ \mathbf{y}(t) &= \mathbf{C} \mathbf{x}(t) + \mathbf{D} \mathbf{u}(t) \end{aligned} \tag{8}$$

Where:

*3.3.1. Reduced Model*

where only some modes of vibration are selected.

18 Advances in Vibration Engineering and Structural Dynamics

1 2

ì ü ï ï ï ï ï ï ï ï ï ï <sup>=</sup> í ý ï ï ï ï ï ï ï ï ï ï î þ

*x x*

...

*x*

*y y*

1 2

*xy*

**Table 9.** Generalized coordinates for full FE model.

**Table 10.** Nodal to Modal transformation.

cated in:

*S*ξ*ixy* =Φ*xy*

**q**

*(t)*

*n*

...

*y*

trices and vector are reported in their modal forms.

*<sup>T</sup> <sup>S</sup>ixy <sup>S</sup>*ξ*<sup>i</sup>* <sup>=</sup>Φ*<sup>z</sup>*

*S*ξ*oxy* =*Soxy*Φ*xy S*ξ*oz* =*Soz*Φ*<sup>z</sup>*

**Table 11.** Input / Output matrices conversion from Nodal to Modal.

*n*

The displacements to modal transformation are reported in Table 10:

Usually a reduced model is used. A typical reduction method is the modal (MK) reduction

1 2 ... *<sup>z</sup>*

(12)

ì ü ï ï ï ï <sup>=</sup> í ý ï ï ï ï î þ

*z*

*<sup>z</sup> (t)*

**q**

*n*

*<sup>T</sup> <sup>S</sup>iz* (14)

*z*

**X-Y Behaviour Z Behaviour**

**X-Y Behaviour Z Behaviour**

**ξ***xy*(*t*)=Φ*xyqxy*(*t*) ξ*z*(*t*)=Φ*zqz*(*t*) (13)

The equations of motion for the reduced model are formerly the same reported in equations (4) to (7), where the nodal displacements is substituted by the modal displacements and ma‐

Input/output selection matrices should be transformed, starting from FEM matrices, as indi‐

**X-Y Behaviour Z Behaviour**


**Table 12.** State Space Representation variables.

## **4. Control design and results**

The aim of this section is to explain the steps followed to perform the suspension control design. A conventional decentralized control strategy is illustrated with two nested loops, the inner for current control and the outer for the position.The detailed description of this strategy is followed by the exposition of an off-line electrical centering strategy which is used equalize electrical parameters of the electromagnets on each actuation stage and allows to get a steady and balanced control action.

#### **4.1. ActiveMagnetic Suspension Control**

Figure 4 shows the classical control strategy for one axis AMB. The system is characterized with a nested control structure, where the inner loops describe the current loops used to achieve a direct actuator's effort drive (force) and the outer loop is used to compensate the rotor position error from the nominal air-gap. Generally, the same strategy is applied for each axis; so they are managed independently from each other and control is called decen‐ tralized. The driving of one axis is performed with two separate H-bridges. To exert a posi‐ tive force on the rotor, current in the upper coil is increased by the control current while the current in the lower coil is decreased by control current and vice versa for negative forces. Also, to linearize the current to force characteristic of an electromagnet a constant bias cur‐ rent is applied to both coils respectively.Position control is performed by using five decen‐ tralized PID. The design and tuning of control laws parameters are well described in [1] and [2]. Here Bode diagram transfer function of position control law is reported (Figure11).

The experimental characterization has been performed by using classical tools of rotordynam‐ ic analysis: Unbalance responses (Figure 12), Waterfalls (Figure 13), orbital tube and orbital view (Figure 14, Figure 15). Theoretical notes on rotodynamic analysis can be found in [29].

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Active Magnetic Bearings offer several technological advantages that make their use manda‐ tory in some particular applications, typically when clean environment is required or main‐ tenance is expensive or difficult to manage. On the other hand, the main drawbacks are mostly the costs, higher than classical ball bearings due to the introduction of sensors and electronic equipment, and the complexity in the design phases of electrical and electronics

One of the aspects where this difficulty is more evident is the centering of the rotating part respect to the stator and in particular to the sensors. Sensors are indeed designed to detect microns of displacements and little inaccuracies in measurements lead to bad working or to

The designer can choice to perform either a geometrical or an electrical centering, depend‐ ing on the priorities of the application requirements. The first (Figure 16.a) consists in putting the rotor at the mechanical center neglecting the electrical differences in electro‐ magnets coils parameters, inductance above all. The latter (Figure 16.b), on the contrary, leads to an equalization of electrical parameters, even if the rotor is not spinning around

**4.2. Off-lineElectricalCentering**

subsystems and control strategies.

system instability in the worst cases.

**Figure 13.** Waterfall plot. a) X1; b) Y1; c) X2; d) Y2.

the geometrical center of the actuators.

**Figure 11.** Control position Bode diagram.

**Figure 12.** Unbalance response. a) Left actuation stage. b) Right actuation stage.

The experimental characterization has been performed by using classical tools of rotordynam‐ ic analysis: Unbalance responses (Figure 12), Waterfalls (Figure 13), orbital tube and orbital view (Figure 14, Figure 15). Theoretical notes on rotodynamic analysis can be found in [29].

## **4.2. Off-lineElectricalCentering**

**4.1. ActiveMagnetic Suspension Control**

20 Advances in Vibration Engineering and Structural Dynamics

**Figure 11.** Control position Bode diagram.

**Figure 12.** Unbalance response. a) Left actuation stage. b) Right actuation stage.

Figure 4 shows the classical control strategy for one axis AMB. The system is characterized with a nested control structure, where the inner loops describe the current loops used to achieve a direct actuator's effort drive (force) and the outer loop is used to compensate the rotor position error from the nominal air-gap. Generally, the same strategy is applied for each axis; so they are managed independently from each other and control is called decen‐ tralized. The driving of one axis is performed with two separate H-bridges. To exert a posi‐ tive force on the rotor, current in the upper coil is increased by the control current while the current in the lower coil is decreased by control current and vice versa for negative forces. Also, to linearize the current to force characteristic of an electromagnet a constant bias cur‐ rent is applied to both coils respectively.Position control is performed by using five decen‐ tralized PID. The design and tuning of control laws parameters are well described in [1] and [2]. Here Bode diagram transfer function of position control law is reported (Figure11).

Active Magnetic Bearings offer several technological advantages that make their use manda‐ tory in some particular applications, typically when clean environment is required or main‐ tenance is expensive or difficult to manage. On the other hand, the main drawbacks are mostly the costs, higher than classical ball bearings due to the introduction of sensors and electronic equipment, and the complexity in the design phases of electrical and electronics subsystems and control strategies.

One of the aspects where this difficulty is more evident is the centering of the rotating part respect to the stator and in particular to the sensors. Sensors are indeed designed to detect microns of displacements and little inaccuracies in measurements lead to bad working or to system instability in the worst cases.

**Figure 13.** Waterfall plot. a) X1; b) Y1; c) X2; d) Y2.

The designer can choice to perform either a geometrical or an electrical centering, depend‐ ing on the priorities of the application requirements. The first (Figure 16.a) consists in putting the rotor at the mechanical center neglecting the electrical differences in electro‐ magnets coils parameters, inductance above all. The latter (Figure 16.b), on the contrary, leads to an equalization of electrical parameters, even if the rotor is not spinning around the geometrical center of the actuators.

**Figure 14.** Orbital tube and orbital view representation on X1Y1 - plane. a) three dimensional view of the tube; b) Projection on the xy-plane; c) and d) projection on the Ωx and Ωy –planes.

**Figure 16.** Mechanical centering (a) vs. Electrical centering (b).

**Figure 17.** Inductance vs. Position.

resistance and inductance values (Table 2) as in Eq. 17:

This behavior is nonlinear as illustrated in Figure 17 and little variation of position and hence on inductance lead to big variation of actuator electrical pole and current control dy‐ namics.Since the electrical dynamics of an electromagnet is depending on supply voltage,

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the electrical pole of the electromagnet is strictly dependent on the distance between the tar‐ get and the actuator as reported in Figure 18. This issue has as consequence a different be‐

(9)

*VDC <sup>E</sup> R Ls* <sup>=</sup> <sup>+</sup>

**Figure 15.** Orbital tube and orbital view on X2Y2 - plane. a) three dimensional view of the tube; b) Projection on the xy-plane; c) and d) projection on the Ωx and Ωy –planes.

In this section an off-line electrical centering technique is exposed. The goal is to make the rotor spins around a point which is not compulsorily coincident with the geometrical center of the actuators but grants the symmetry of the electrical parameters of them. It is well known that the inductance value of an electromagnet is depending on the distance between the ferromagnetic target (the rotor in this case) and the electromagnet itself.

Rotors on Active Magnetic Bearings: Modeling and Control Techniques http://dx.doi.org/10.5772/51298 23

**Figure 16.** Mechanical centering (a) vs. Electrical centering (b).

**Figure 17.** Inductance vs. Position.

**Figure 14.** Orbital tube and orbital view representation on X1Y1 - plane. a) three dimensional view of the tube; b)

**Figure 15.** Orbital tube and orbital view on X2Y2 - plane. a) three dimensional view of the tube; b) Projection on the

In this section an off-line electrical centering technique is exposed. The goal is to make the rotor spins around a point which is not compulsorily coincident with the geometrical center of the actuators but grants the symmetry of the electrical parameters of them. It is well known that the inductance value of an electromagnet is depending on the distance between

the ferromagnetic target (the rotor in this case) and the electromagnet itself.

Projection on the xy-plane; c) and d) projection on the Ωx and Ωy –planes.

22 Advances in Vibration Engineering and Structural Dynamics

xy-plane; c) and d) projection on the Ωx and Ωy –planes.

This behavior is nonlinear as illustrated in Figure 17 and little variation of position and hence on inductance lead to big variation of actuator electrical pole and current control dy‐ namics.Since the electrical dynamics of an electromagnet is depending on supply voltage, resistance and inductance values (Table 2) as in Eq. 17:

$$E = \frac{VDC}{R + Ls} \tag{9}$$

the electrical pole of the electromagnet is strictly dependent on the distance between the tar‐ get and the actuator as reported in Figure 18. This issue has as consequence a different be‐ havior of closed loop current control as illustrated in Figure 19 (a and b). It can be noticed that the same current control applied to the two opposite electromagnets of the same actua‐ tion axis without electrical centering generates two different closed loop responses. Few mi‐ crons of Airgap generates differences of hundreds of Hertz on current control bandwidth.

Considering that this behavior is generated by a difference of inductance value of the two electromagnets, by acting on the position reference with offset corrections of the outer po‐ sition control, the rotor can be set to spin around a point where electrical parameters are equalized and current loop bandwidths of both the electromagnets are the same (Figure 19 (c and d)). Further studies and research are being conducted on this strategy since this process can be performed in an on-line automatic routine with an adaptive technique, able to change the control parameters of the inner current loop while the Airgap is chang‐ ing, i.e. when the rotor is oscillating.

**Figure 19.** Electrical centering results. a) X- current loop Bode diagram before centering; b) X+ current loop Bode dia‐

, Mario Silvagni and Lester D. Suarez

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[1] Bleuler, H., Cole, M., Keogh, P., Larsonneur, R., Maslen, E., Nordmann, R., Okada, Y., Schweitzer, G., & Traxler, A. (2009). *Magnetic Bearings Theory, Design, and Applica‐*

[2] Chiba, A., Fukao, T., Ichikawa, O., Oshima, M., Takemoto, M., & Dorrell, D. G.

gram before centering; c, d) X-/X+ current loop Bode diagrams after centering.

\*Address all correspondence to: angelo.bonfitto@polito.it

Mechanics Department, Mechatronics Laboratory – Politecnico di Torino, Italy

*tion to Rotating Machinery*, Springer, Berlin Heidelberg.

(2005). *Magnetic Bearings and Bearingless drives*, Elsevier, Oxfor.

**Author details**

**References**

Andrea Tonoli, Angelo Bonfitto\*

**Figure 18.** Electrical pole trend at varying of inductance value.

## **5. Conclusions**

In this chapter the modeling, the design and the experimental tests phases of a rotor equip‐ ped with active magnetic bearings have been described. The description deals with rotordy‐ namic aspects as well as electrical, electronic and control strategies subsystem. The control design of a standard decentralized SISO strategy and the details of an innovative off-line electrical centering technique have been exposed.Experimental results have been exposed highlighting mainly rotordynamics and control aspects.

**Figure 19.** Electrical centering results. a) X- current loop Bode diagram before centering; b) X+ current loop Bode dia‐ gram before centering; c, d) X-/X+ current loop Bode diagrams after centering.
