**5. Results of the investigations**

The model test stand of the flexible power-transmission shaft is used to carry out the experiment whose aim is to verify the identification procedure of dynamic coefficients of the active magnetic bearing (Figure 16). The test rig consists of the horizontal shaft supported in two rolling bearings mounted at both ends. It is driven by an electric motor connected to the shaft through an elastic membrane coupling with smooth rotation control and fed by a frequency converter. An active magnetic bearing operates as an auxiliary bearing that modifies the dynamic properties of the shaft line. Between the magnetic bearing and the shaft right end, there is a rigid disk which allows one to mount balancing weights for the real structure. The mass of the rotating system is equal to *4.85 kg*, the shaft line length equals *1923 mm.* The test stand allows one to investigate the effects of the magnetic bearing on dynamic properties (vibration level, displacement and the coefficient of vibration amplification of subsequent critical frequencies) and to control vibrations of the long, flexible rotor. A kinematics exciter is fixed on the disk, on which masses of test unbalancing can be mounted. After introducing a selected program for magnetic bearing control, harmonic vibrations of the shaft of the frequency *(10, 20, 30, … 80) Hz* and the assigned amplitude were excited. For each frequency under analysis, the time histories of displacements and currents in the magnetic bearing, which were subject to respective calculation procedures, were recorded, and then the bearing dynamic parameters were estimated. To conduct the measurement and calculation procedures, a measurement system with *DBK 15* input systems made by *IOtech,* operating with a PC and employing the *Daq/112B* type *PCMCIA* measurement card of the resolution equal to *12 bites* and the maximum sampling frequency of *100kHz*, was applied.

**Figure 16.** Configuration of the test rig

180 Performance Evaluation of Bearings

selected part of the time history.

coefficients *CXX* in such a way as to make the sum of squares of differences minimal for the

[ N ] FXmag X\*Kxx+VX\*Cxx

**Figure 15.** Measured magnetic response component along the control axis *X* - *FXmag* and its modelled

In the method of identification of bearing dynamic coefficients, it is required that the theoretically calculated magnetic response force is the closest approximation of its function obtained in the measurements and that the share of synchronous components in the curves

0.05 0.10 0.15 0.20 0.25

The model test stand of the flexible power-transmission shaft is used to carry out the experiment whose aim is to verify the identification procedure of dynamic coefficients of the active magnetic bearing (Figure 16). The test rig consists of the horizontal shaft supported in two rolling bearings mounted at both ends. It is driven by an electric motor connected to the shaft through an elastic membrane coupling with smooth rotation control and fed by a frequency converter. An active magnetic bearing operates as an auxiliary bearing that modifies the dynamic properties of the shaft line. Between the magnetic bearing and the shaft right end, there is a rigid disk which allows one to mount balancing weights for the real structure. The mass of the rotating system is equal to *4.85 kg*, the shaft line length equals *1923 mm.* The test stand allows one to investigate the effects of the magnetic bearing on dynamic properties (vibration level, displacement and the coefficient of vibration amplification of subsequent critical frequencies) and to control vibrations of the long, flexible rotor. A kinematics exciter is fixed on the disk, on which masses of test unbalancing can be mounted. After introducing a selected program for magnetic bearing control,

amplitude were excited. For each frequency under analysis, the time histories of displacements and currents in the magnetic bearing, which were subject to respective calculation procedures, were recorded, and then the bearing dynamic parameters were

 *80) Hz* and the assigned

[ s ]

time history *FX lin* with the identified dynamic coefficients *KXX , CXX*

**5. Results of the investigations** 


of displacement, current and magnetic response force is dominant [5,6,9].

harmonic vibrations of the shaft of the frequency *(10, 20, 30, …* 

The voltage time histories corresponding to displacements (positions) of the journal along both the control axes *X,Y* were recorded *on-line* on respective inputs of the measurementcontrol module. These were two voltage signals *0-24V* from *Bently-Nevada* type *3300* eddycurrent transducers of relative vibrations*.* The voltage time histories corresponding to currents flowing in electromagnets were measured and recorded. These were four voltage signals *0-5V* from current-voltage *LEM* type transducers.

The DaqView v.7.9.8 software was used for recording purposes. There were *4000* measurements made, at the sampling frequency of *8kHz/channel*. The results were stored in binary files of the data acquisition system, and then converted into text files. The programs for analysis of dynamics and identification procedures of bearing dynamic parameters, according to the methodology proposed, were developed with the *MS Excel* spreadsheet.

Exemplary time histories of the quantities measured are shown in Figures 17 and 18 and of those calculated - in Figures 19 and 20 for the magnetic bearing of the selected configuration of the control program, at the kinematic excitation of the frequency *40 Hz* and the assigned amplitude, whose value was such as to obtain the dominant share of synchronous components in the time histories under analysis and to obtain the linear range of magnetic response forces.

**Figure 17.** Displacements for both the control axes *X,Y* and the shaft motion trajectory

The occasional disturbances which occur in the recorded time histories of displacements (Figure 17) are amplified by the digital differentiation and the effect of these disturbances is very distinct in the time history of the velocity component *VY* (Figure 19a). This does not affect the calculation accuracy, where the characteristics are approximated with the harmonic time history.

Theoretical and Experimental Investigations of

 *CYY*. The analysis of this state showed that for

0 20 40 60 80

 Cxx Cyy

**[ Hz ]**

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 183

bearing dynamic coefficients, namely: the stiffness coefficients *KXX , KYY* and the damping coefficients *CXX , CYY*, as a function of frequency at the given excitation frequency and the

**Figure 21.** a. – Stiffness *KXX* , *KYY* versus frequency b. – Damping *CXX* , *CYY* versus frequency

**[ Hz ]**

 *KYY , CXX*

conducted with the simulation model of the magnetic bearing.

shaft line with supports of identified dynamic properties.

**6. Modification of the dynamics of the rotating system** 

It means that the start-up and shut-down characteristics of the model shaft line under such magnetic bearing operating conditions should be the overcritical characteristics. The experimentally determined dynamic coefficients of the bearing show some anisotropy of the

0.0E+0

2.0E+2

4.0E+2

6.0E+2

8.0E+2

**[Ns/m]**

isotropic properties of the energy transmission systems (symmetrical saturation–control current characteristics), this scattering resulted from the scattering of constant values of electromagnets for the given axis. This conclusion was confirmed by the calculations

The experiment was conducted for various configurations of the program controlling the auxiliary magnetic support used in the model shaft line system. The determined dynamic coefficients were employed in subsequent stages of the investigations in modelling and numerical calculations of start-up and shut-down characteristics of the flexible powertransmission shaft. The generated numerical characteristics were verified experimentally through a comparison to the start-up and shut-down curves recorded for the real model

The proposed methodology of measurement of response and dynamic coefficients of the magnetic bearing is a very important tool in designing dynamics and vibration control of machine rotors in which active magnetic bearings are applied. It allows one to find analogies to classical bearing systems and to employ professional calculation codes for evaluation of the effects of modification in the dynamic properties of shaft lines introduced through changes in the configuration of the program controlling its active magnetic

The last step of this work is to present the experimental results of maintaining a low level of vibrations in the whole operating range of the flexible rotor by the application of an

given configuration of the control program (Figures 21).

*a b* 

properties for individual axes *KXX*

0 20 40 60 80

 Kxx Kyy

0.0E+0

2.0E+5

4.0E+5

6.0E+5

8.0E+5

**[ N/m ]**

supports [4-9].

**Figure 18.** Time histories of the currents in electromagnet windings *IXT* ,*IXB , IYT* ,*IYB* - averaged

**Figure 19.** a. – Velocity components for both the control axes *VX,VY* – averaged b. – Variable gaps for individual electromagnets *XT* ,*sXB , sYT* ,*sYB*

**Figure 20.** a. – Measured magnetic response component *FXac* and the force *FXacT = FX lin* modelled with the identified dynamic coefficients *KXX , CXX* 

b. – Components of the magnetic response related to the stiffness *FXstiff* and to the damping *FXdamp*

The measurement and calculation cycles were conducted for various excitation frequencies in the range *(10 80)Hz*, which allowed one to build functions representing the values of the bearing dynamic coefficients, namely: the stiffness coefficients *KXX , KYY* and the damping coefficients *CXX , CYY*, as a function of frequency at the given excitation frequency and the given configuration of the control program (Figures 21).

182 Performance Evaluation of Bearings

harmonic time history.

very distinct in the time history of the velocity component *VY* (Figure 19a). This does not affect the calculation accuracy, where the characteristics are approximated with the

**[ A ] IXTav5 IYTav5**

 **IXBav5 IYBav5**

**Figure 18.** Time histories of the currents in electromagnet windings *IXT* ,*IXB , IYT* ,*IYB* - averaged

 **SXB SYB** *a b* 

**[ s ]**

0.20 0.22 0.24 0.26 0.28 0.30

**[ s ]**

0.20 0.22 0.24 0.26 0.28 0.30

0.20 0.22 0.24 0.26 0.28 0.30

**[ m ] SXT SYT**

**[ s ]**

**[ s ]**

**Figure 19.** a. – Velocity components for both the control axes *VX,VY* – averaged

**[ s ]**

**Figure 20.** a. – Measured magnetic response component *FXac* and the force *FXacT = FX lin* modelled with the

**[ N ] FXstiff FXdamp** *a b* 

The measurement and calculation cycles were conducted for various excitation frequencies

 *80)Hz*, which allowed one to build functions representing the values of the




10

30

50

0.0000 0.0002

0.0004 0.0006

0.0008 0.0010

b. – Components of the magnetic response related to the stiffness *FXstiff* and to the damping *FXdamp*

b. – Variable gaps for individual electromagnets *XT* ,*sXB , sYT* ,*sYB*

0,20 0,22 0,24 0,26 0,28 0,30

**[ N ] FXac FXacT**

0.20 0.22 0.24 0.26 0.28 0.30

**[mm/s] VXav3 VYav3**

0.0

0.4

0.8

1.2

1.6

identified dynamic coefficients *KXX , CXX* 

in the range *(10* 




10

30

50


**Figure 21.** a. – Stiffness *KXX* , *KYY* versus frequency b. – Damping *CXX* , *CYY* versus frequency

It means that the start-up and shut-down characteristics of the model shaft line under such magnetic bearing operating conditions should be the overcritical characteristics. The experimentally determined dynamic coefficients of the bearing show some anisotropy of the properties for individual axes *KXX KYY , CXX CYY*. The analysis of this state showed that for isotropic properties of the energy transmission systems (symmetrical saturation–control current characteristics), this scattering resulted from the scattering of constant values of electromagnets for the given axis. This conclusion was confirmed by the calculations conducted with the simulation model of the magnetic bearing.

The experiment was conducted for various configurations of the program controlling the auxiliary magnetic support used in the model shaft line system. The determined dynamic coefficients were employed in subsequent stages of the investigations in modelling and numerical calculations of start-up and shut-down characteristics of the flexible powertransmission shaft. The generated numerical characteristics were verified experimentally through a comparison to the start-up and shut-down curves recorded for the real model shaft line with supports of identified dynamic properties.

The proposed methodology of measurement of response and dynamic coefficients of the magnetic bearing is a very important tool in designing dynamics and vibration control of machine rotors in which active magnetic bearings are applied. It allows one to find analogies to classical bearing systems and to employ professional calculation codes for evaluation of the effects of modification in the dynamic properties of shaft lines introduced through changes in the configuration of the program controlling its active magnetic supports [4-9].
