**3.3. Numerical calculations and verifications**

172 Performance Evaluation of Bearings

programmed correction.

0

1

2

3

4

5

*I [A]*

0

power amplifier supply voltage

1

2

3

4

*I [ A ]*

0 10 20 30 40 50 60

result, they render the asymmetry of mechanical properties of the system for these axes

*a. b.* 

**Figure 6.** a. - Experimental *pulse-width modulation W - control current I* characteristics of the bearing actuator systems before programmed correction, b. - Theoretical and experimental characteristics after

 *W [ % ]*

systems for each pair of electromagnets and for each bearing control axis *y* and *x*.

*EM3, L = 80 mH , R = 3,1* 

 *fPWM = 1667 Hz*

For the given parameters of a real bearing system, there is always a scattering of properties of the actuating system for each pair of electromagnets. The developed procedure enables one to tune these characteristics in order to obtain the symmetry of operation of actuating

0

1

2

3

4

*I [A]*

An alternative to compensation for the asymmetry of bearing actuator properties by designing a proper structure and algorithm of the controller is to develop a method of its programmed correction. On the assumption that the source of the identified asymmetry lies in a scatter of actuator parameters, a correction method that leads to minimisation of the effect of the scatter of their characteristics for each bearing control axis has been proposed.

 *Uz = 80 V*

 *Uz = 50 V*

 *Uz = 30 V*

0 10 20 30 40 50 60

characteristics of the bearing actuator systems after the programmed correction for various values of the

**Figure 7.** Theoretical (solid line) and experimental (dots) *pulse-width modulation - control current*

W [ % ]

0 10 20 30 40 50 60

 *Exp. - axis y (-2,45% corr.) Exp. - axis x (-5,40% corr.)*

 *Calculations*

*W [ % ]*

when the feedback loop is closed and the bearing control system is turned on.

*Axis y*

*Axis x*

In the analysis of model investigation results, the same quantities have been chosen as those recorded in a real magnetic bearing system, namely: pulse-width modulation, displacement, static equilibrium position, journal trajectory, phase portrait, Bode's plot, etc. It has enabled their direct verification with the experiment. The verification has been carried out on the test stand with the built radial bearing, whose part is presented in Figure 8.

**Figure 8.** Structure of the built radial bearing

The investigations of the bearing dynamics comprise three basic modes of its operation:

Theoretical and Experimental Investigations of

0.0 0.1 0.2 0.3 0.4 0.5

EM3

[ A ] EM4

time [ s ]

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 175

modulation *Wfb* is realized through the application of an additional feedback loop with

In Figs. 8 a and b changes in the pulse-width modulation as a function of time in the mode of suspension of the journal in the bush for the windings of the top electromagnet *EM4* and the bottom one *EM3* interacting with the axis x are shown. The data recording for the real bearing structure was carried out by means of the developed *PRDP* procedure. The presented calculation results and the recorded results refer to the same parameters of the controller. A good convergence of the results has been obtained, which confirms reliability

On the basis of the calculated pulse-width modulations, the bearing actuators generate currents that control the windings of corresponding bush electromagnets. The experimentally verified procedures that represent the real actuator operation for the

assigned parameters have been employed in the algorithm of the model operation.

**Figure 10.** Currents in the windings of the top *EM4* and bottom *EM3* electromagnet - axis *x*

time [ s ]

For the real structure, the values of correction, indispensable to obtain the symmetry of bearing properties, have been introduced for each pair of windings for both the control axes

0.0

0.5

1.0

1.5

current

In Figures 10 a and b changes in the current versus time are shown in the mode of suspension of the journal in the bush for the electromagnet top windings *EM4, EM1* and bottom windings *EM3, EM2*, interacting with the corresponding control axes *x* and *y*. The control currents in the bush top and bottom winding along each control axis are timevariable and are a function of the change in the journal position with respect to the bush

Figures 11a and b show changes in the displacement of the magnetic bearing journal at the moment an active magnetic bearing is activated for both the control axes *x, y* for the real

In Figures 11 and 12 an effect of the bearing system operation in its first operation mode in the form of the calculated and recorded changes in the position of the journal with respect to

a. - numerical simulations, b. - experimental results

0.0 0.1 0.2 0.3 0.4 0.5

EM3

*a b* 

[ A ] EM4

until the equilibrium position is achieved.

system and its simulation model, respectively.

a possibility of programmed assigning the value of the feedback coefficient *Kfb*.

of the model built.

*x* and *y*.

0.0

0.5

1.0

1.5

current


Some selected test results for the first mode of the system operation are presented.

The nominal values of parameters that have been assumed in the model calculations are as follows:


The bearing constant *K*, as well as the power amplifier circuit resistance and inductance, *R, L*, have been determined experimentally through suitable measurement procedures for the real structure of the bearing system, whose actuators operate at the above-mentioned values of the supply voltage *UZ* and the frequency *fPWM* .

In order to present the diagnostic capabilities of the model developed and to indicate a convergence of its operation with the real system, the suitably selected characteristics calculated numerically for the defined nominal parameters and those recorded for the real bearing system have been shown in Figures 9 - 13.

**Figure 9.** Changes in the pulse-width modulation during the suspension of the journal in the bush for the control axis x (electromagnets: top - *EM4* and bottom - *EM3*) – a. - numerical simulations, b.– experimental results

For one control axis during given sampling time, pulses with the pulse-width modulation *W* proportional to the signal produced by the controller are calculated on the basis of the displacement measurement of the journal with respect to the bush. After its comparison with the assigned pulse-width modulation that determines the bearing *WB* operating point and the pulse-width modulation that corresponds to the value of the current flowing in the top and bottom electromagnet winding *Wfb* , the pulse-width modulation *Wfb* is realized through the application of an additional feedback loop with a possibility of programmed assigning the value of the feedback coefficient *Kfb*.

174 Performance Evaluation of Bearings


follows:

experimental results

0

20

40

60

80

100

W [ % ]


power amplifier supply voltage *UZ = 80V*

bearing constant *K= 2.4 10 –5 Nm2/A2*

frequency of control pulses *fPWM =1667Hz TPWM= 600*

inductance of the electromagnet winding *L = 80mH*

of the supply voltage *UZ* and the frequency *fPWM* .

bearing system have been shown in Figures 9 - 13.

*a b* 

 EM4 EM3

0.00 0.02 0.04 0.06 0.08 0.10


The investigations of the bearing dynamics comprise three basic modes of its operation:

Some selected test results for the first mode of the system operation are presented.

resistance of the power amplifier circuit, including the windings *R = 3.1*

The nominal values of parameters that have been assumed in the model calculations are as

The bearing constant *K*, as well as the power amplifier circuit resistance and inductance, *R, L*, have been determined experimentally through suitable measurement procedures for the real structure of the bearing system, whose actuators operate at the above-mentioned values

In order to present the diagnostic capabilities of the model developed and to indicate a convergence of its operation with the real system, the suitably selected characteristics calculated numerically for the defined nominal parameters and those recorded for the real

**Figure 9.** Changes in the pulse-width modulation during the suspension of the journal in the bush for the control axis x (electromagnets: top - *EM4* and bottom - *EM3*) – a. - numerical simulations, b.–

0

20

40

60

80

100

W [%]

time [ s ]

For one control axis during given sampling time, pulses with the pulse-width modulation *W* proportional to the signal produced by the controller are calculated on the basis of the displacement measurement of the journal with respect to the bush. After its comparison with the assigned pulse-width modulation that determines the bearing *WB* operating point and the pulse-width modulation that corresponds to the value of the current flowing in the top and bottom electromagnet winding *Wfb* , the pulse-width

*s*

 EM4 EM3

0.00 0.02 0.04 0.06 0.08 0.10

time [ s ]

In Figs. 8 a and b changes in the pulse-width modulation as a function of time in the mode of suspension of the journal in the bush for the windings of the top electromagnet *EM4* and the bottom one *EM3* interacting with the axis x are shown. The data recording for the real bearing structure was carried out by means of the developed *PRDP* procedure. The presented calculation results and the recorded results refer to the same parameters of the controller. A good convergence of the results has been obtained, which confirms reliability of the model built.

On the basis of the calculated pulse-width modulations, the bearing actuators generate currents that control the windings of corresponding bush electromagnets. The experimentally verified procedures that represent the real actuator operation for the assigned parameters have been employed in the algorithm of the model operation.

**Figure 10.** Currents in the windings of the top *EM4* and bottom *EM3* electromagnet - axis *x* a. - numerical simulations, b. - experimental results

For the real structure, the values of correction, indispensable to obtain the symmetry of bearing properties, have been introduced for each pair of windings for both the control axes *x* and *y*.

In Figures 10 a and b changes in the current versus time are shown in the mode of suspension of the journal in the bush for the electromagnet top windings *EM4, EM1* and bottom windings *EM3, EM2*, interacting with the corresponding control axes *x* and *y*. The control currents in the bush top and bottom winding along each control axis are timevariable and are a function of the change in the journal position with respect to the bush until the equilibrium position is achieved.

Figures 11a and b show changes in the displacement of the magnetic bearing journal at the moment an active magnetic bearing is activated for both the control axes *x, y* for the real system and its simulation model, respectively.

In Figures 11 and 12 an effect of the bearing system operation in its first operation mode in the form of the calculated and recorded changes in the position of the journal with respect to the bush as a function of time is presented for both the control axes and for the orbit composed of these displacements.

Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 177

time [ s ]

**Figure 13.** Magnetic force *Fymag* generated in the bearing system - axis *y*

0

10

20

30

40

50

60

Fmy [ N ]

model and real bearing.

[8,9].

that can occur at the system prototype start-up.

The agreement between the simulation investigations and the real bearing structure investigations has confirmed that the simulation model developed is a reliable tool that can be employed in designing a bearing system and in forecasting the tendencies of changes in

0.0 0.1 0.2 0.3 0.4 0.5

Their comparison indicates the feasibility of the simulation model in practical applications.

In order to apply simulation investigations in practice, their verification for an actual object is needed. The basic assumptions connected with the developed simulation model of an active magnetic bearing system are presented. The simulation and experimental investigations have been verified on the test stand for a digitally controlled journal active magnetic bearing. The results discussed confirm a convergence between the operation of the

A reliable theoretical model that allows for analysis of the bearing dynamics under hypothetical, extreme loads reduces the designing time and enables one to minimize errors

The tool allows for testing different variants of the magnetic bearing system operation, which has been confirmed by the experimental investigations conducted in a wide range. A diagnostic capability of the non-linear numerical model of the magnetic bearing allows to develop a method for the identification of dynamical parameters: stiffness and damping

**4. Concept of the method for the identification of dynamic parameters** 

In the proposed measurement method, a dependence of the vector of the resultant magnetic force acting on the machine shaft as a function of the journal position in the magnetic radial bearing and the currents that flow through the windings of actuators (electromagnets) is employed. The magnetic bearing response vector is a sum of the forces generated by bearing

its operation under the assigned level of disturbances and assigned excitations.

**Figure 11.** Displacement of the rotor a. - numerical simulation, b.- experimental results

**Figure 12.** Orbit of the rotor a. - numerical simulations, b. - experimental results

At the ideal symmetry of the properties of the actuators for each control axis, which is assumed in the model, symmetrical time histories of the journal displacement for both the position control axes *x(t) = y(t)* are obtained - Figure10 a. In the real system, slight differences in the time history of the journal displacement occur at the moment of its startup in comparison with the model. These differences result from: non-ideal geometry of the journal, asymmetry of the arrangement of the position sensors with respect to the journal or asymmetry of the journal position with respect to the bush (Figures 11b, 12b).

The character of the changes in the journal position as well as the time that passes from the moment the control system is activated up to the moment in which stable suspension of the mass supported in the magnetic bearing is achieved are the same for the model and for the real structure under investigation.

In Figure 13 changes in the magnetic force generated in the bearing system during suspension of the journal in the bush versus time are shown. The characteristics is obtained on the basis of the simulations.

Theoretical and Experimental Investigations of Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 177

**Figure 13.** Magnetic force *Fymag* generated in the bearing system - axis *y*

176 Performance Evaluation of Bearings



0

200

400

composed of these displacements.

[m] X

0.0 0.1 0.2 0.3 0.4 0.5

*a b* 

Y

real structure under investigation.



0


250

500

*a b* 

X - Y [ m ]

L bezp.

on the basis of the simulations.

the bush as a function of time is presented for both the control axes and for the orbit



0

0.0 0.1 0.2 0.3 0.4 0.5



0


250

500

[ m ] X - Y

L bezp.

Y

[m] X

Time [s]

[ m ]

200

400

**Figure 11.** Displacement of the rotor a. - numerical simulation, b.- experimental results

[ m ]

Time [s]

**Figure 12.** Orbit of the rotor a. - numerical simulations, b. - experimental results

asymmetry of the journal position with respect to the bush (Figures 11b, 12b).

At the ideal symmetry of the properties of the actuators for each control axis, which is assumed in the model, symmetrical time histories of the journal displacement for both the position control axes *x(t) = y(t)* are obtained - Figure10 a. In the real system, slight differences in the time history of the journal displacement occur at the moment of its startup in comparison with the model. These differences result from: non-ideal geometry of the journal, asymmetry of the arrangement of the position sensors with respect to the journal or

The character of the changes in the journal position as well as the time that passes from the moment the control system is activated up to the moment in which stable suspension of the mass supported in the magnetic bearing is achieved are the same for the model and for the

In Figure 13 changes in the magnetic force generated in the bearing system during suspension of the journal in the bush versus time are shown. The characteristics is obtained The agreement between the simulation investigations and the real bearing structure investigations has confirmed that the simulation model developed is a reliable tool that can be employed in designing a bearing system and in forecasting the tendencies of changes in its operation under the assigned level of disturbances and assigned excitations.

Their comparison indicates the feasibility of the simulation model in practical applications.

In order to apply simulation investigations in practice, their verification for an actual object is needed. The basic assumptions connected with the developed simulation model of an active magnetic bearing system are presented. The simulation and experimental investigations have been verified on the test stand for a digitally controlled journal active magnetic bearing. The results discussed confirm a convergence between the operation of the model and real bearing.

A reliable theoretical model that allows for analysis of the bearing dynamics under hypothetical, extreme loads reduces the designing time and enables one to minimize errors that can occur at the system prototype start-up.

The tool allows for testing different variants of the magnetic bearing system operation, which has been confirmed by the experimental investigations conducted in a wide range. A diagnostic capability of the non-linear numerical model of the magnetic bearing allows to develop a method for the identification of dynamical parameters: stiffness and damping [8,9].
