**7. Numerical calculations and verifications**

184 Performance Evaluation of Bearings

rotor occurs.

auxiliary active magnetic bearing where the biggest anticipated dynamic deflection of the

The application of an auxiliary magnetic bearing to modify the dynamics of the rotating system demands the comparison of the classical (without a magnetic bearing) and modified design. In the proposed construction of the test stand, a flexible rotor is supported in ball

> *Digitally controlled magnetic bearing*

bearings. An electric engine integrated with the shaft is used as a drive (Figure 22).

**Figure 22.** Scheme of the rotating system with an auxiliary active magnetic bearing

designed magnetic bearings are as follows:

diameter of the journal 89.5 mm

number of electromagnet coil pairs4

unbalance occurring in the centre of the rotor.

two axes of control *x, y* with eddy-current transducers.

radial clearance 0.5 mm

 length 45.9 mm static load 300 N

The length and the diameter of the shaft is 700 mm and 17 mm, respectively. Each bearing support is connected to the foundation by rigid elements. The mechanical parameters of the

In this design, the feasible assembly and disassembly of the auxiliary magnetic bearing is assumed. It is the position of the auxiliary magnetic bearing, mounted on the shaft of the test stand, which allows one to model the mode of lateral vibrations. A numerical experiment shows the usefulness of this concept in the case of the test stand rotor. However, the practical application of the presented idea demands serious experimental verification. The aim of this work is to present the results of the experimental investigations of the dynamic response of the system to the synchronous excitation corresponding to the The test stand as presented in Figure 22 was modelled numerically. A professional program *DYROBES* that allows for modelling the dynamics of the shaft line of rotary machines was employed in the numerical calculations. Numerical calculation methods allow one to carry out a complete analysis of machine rotor vibrations. The geometry of the rotating system was modelled with discrete elements. The magnetic bearing journal is situated in the centre of the shaft line (Figure 23).

In Figures 23a and b a shaft line model and calculated modes of critical frequencies of the rotating system for two configurations, i.e. when the magnetic bearing is turned off and when it provides an additional support of the stand, are depicted.

In the configuration under consideration, the dynamic properties of the magnetic bearing are a vital element that decides about a value of the critical frequency of the analysed shaft line. These properties are connected with the assumed parameters of the control system and were identified through the analysis of the bearing simulation model.

The analysis of the rotating system dynamics was performed and the first lateral critical frequency and its respective deflection line of the rotor supported in ball bearings were determined *(38.2Hz)*. The required value of the dynamic stiffness of the magnetic bearing that allows for avoiding the necessity of exceeding the critical value at the start-up of the model shaft line is equal to *KM 3E+5 N/m*.

The second stage of calculations consisted in a determination of the theoretical start-up curve of the modelled shaft line of the test stand. After introducing the linearized coefficients of stiffness and damping of the auxiliary active magnetic bearing, determined on the basis of the model, a response to the rotating synchronous excitation was determined.

Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 187

 *0.5s* a.) auxiliary magnetic bearing turned off, b.)

 *200Hz* for the fixed rotor

**Figure 24.** Time history of the response (displacement) recorded along the direction *x* for the fixed rotor

To show the dynamics of the response of the system with the activated magnetic bearing in Figure 24b, the same time histories of the response are presented for the shorter recording

During this time, an effect of the system modification that consists in an activation of the additional magnetic bearing with the programmable assigned dynamic properties is clearly

In Figure 25 the transmittances of the rotating system measured on the magnetic bearing

The analysis of these functions shows that in the system with the bearing turned off, weakly damped vibrations with the frequency of approximately *n = 39 Hz* (the first vibration

*A* - recording time *t = 0* 

time.

auxiliary magnetic bearing turned on

journal have been compared.

*4s*, *B* - recording time *t = 0* 

visible. Then, the shaft vibrations disappear after approximately *t = 0.05s*.

**Figure 25.** Frequency response for the axis *x* in the frequency range *n = 0* 

frequency of the rotor in this configuration - Figure 26a) dominate.

a) auxiliary magnetic bearing turned off, b) auxiliary magnetic bearing turned on

**Figure 23.** Shaft line model of the test stand and the first critical mode of vibrations a) magnetic bearing turned off, b.) magnetic bearing turned on

The synchronous excitation with a force coming from the residual unbalance was introduced. The values of the vibration amplitude as a function of frequency are the system response. For the first critical frequency, the vibration amplitude reaches the value of *260m p-p* for the rotating system supported only in ball bearings (Figure 26b). After introducing the auxiliary magnetic bearing in the form of dynamic coefficients of stiffness and damping into the model calculations, the theoretical Bode plot shown in Figure 26 was obtained.

The experimental investigations, like the theoretical ones, were carried out in two stages:


In the first stage, the dynamics of the rotating system was analysed by means of the modal testing method.

For a fixed shaft of the test stand, the tests were carried out for two variants of operation: with the magnetic bearing turned on and turned off.

The system was forced to vibrate by means of the excitation pulse force, whose direction was convergent with the direction of the response measurement (selected control axis). The spectral concentration of this excitation in the frequency range investigated *0200 Hz* was practically constant.

The analysis of the time histories (Figure 24a) points out to the fact that in the system with the magnetic bearing turned off, weakly damped oscillations occur as a result of the excitation and they disappear after approximately *t = 3s*.

In the system in which the bearing is activated, these vibrations are strongly damped. In the whole time span observed, slight oscillations of the rotor occur. They result from the principle of bearing operation, which consists in the continuous control of the assumed position of the rotor balance. The time histories were recorded in the time interval *t = 4s* in order to provide evidence of stable operation of the system after the pulse excitation force had been applied.

Theoretical and Experimental Investigations of Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 187

186 Performance Evaluation of Bearings

of *260*

was obtained.

testing method.

practically constant.

had been applied.


with the magnetic bearing turned on and turned off.

excitation and they disappear after approximately *t = 3s*.

coefficients of stiffness and damping of the auxiliary active magnetic bearing, determined on the basis of the model, a response to the rotating synchronous excitation was determined.

The synchronous excitation with a force coming from the residual unbalance was introduced. The values of the vibration amplitude as a function of frequency are the system response. For the first critical frequency, the vibration amplitude reaches the value

introducing the auxiliary magnetic bearing in the form of dynamic coefficients of stiffness and damping into the model calculations, the theoretical Bode plot shown in Figure 26

The experimental investigations, like the theoretical ones, were carried out in two stages:

In the first stage, the dynamics of the rotating system was analysed by means of the modal

For a fixed shaft of the test stand, the tests were carried out for two variants of operation:

The system was forced to vibrate by means of the excitation pulse force, whose direction was convergent with the direction of the response measurement (selected control axis). The

The analysis of the time histories (Figure 24a) points out to the fact that in the system with the magnetic bearing turned off, weakly damped oscillations occur as a result of the

In the system in which the bearing is activated, these vibrations are strongly damped. In the whole time span observed, slight oscillations of the rotor occur. They result from the principle of bearing operation, which consists in the continuous control of the assumed position of the rotor balance. The time histories were recorded in the time interval *t = 4s* in order to provide evidence of stable operation of the system after the pulse excitation force

*200 Hz* was

spectral concentration of this excitation in the frequency range investigated *0*

*m p-p* for the rotating system supported only in ball bearings (Figure 26b). After

**Figure 23.** Shaft line model of the test stand and the first critical mode of vibrations


a) magnetic bearing turned off, b.) magnetic bearing turned on

**Figure 24.** Time history of the response (displacement) recorded along the direction *x* for the fixed rotor *A* - recording time *t = 0 4s*, *B* - recording time *t = 0 0.5s* a.) auxiliary magnetic bearing turned off, b.) auxiliary magnetic bearing turned on

To show the dynamics of the response of the system with the activated magnetic bearing in Figure 24b, the same time histories of the response are presented for the shorter recording time.

During this time, an effect of the system modification that consists in an activation of the additional magnetic bearing with the programmable assigned dynamic properties is clearly visible. Then, the shaft vibrations disappear after approximately *t = 0.05s*.

**Figure 25.** Frequency response for the axis *x* in the frequency range *n = 0 200Hz* for the fixed rotor a) auxiliary magnetic bearing turned off, b) auxiliary magnetic bearing turned on

In Figure 25 the transmittances of the rotating system measured on the magnetic bearing journal have been compared.

The analysis of these functions shows that in the system with the bearing turned off, weakly damped vibrations with the frequency of approximately *n = 39 Hz* (the first vibration frequency of the rotor in this configuration - Figure 26a) dominate.

The transmittance of the system with the bearing turned on exhibits strong vibration damping. The applied pulse excitation is not able to generate self-vibrations of the rotor. The recorded frequency response (Figure 25b) is located on the level of noises and equals *6m*. The first frequency of self-vibrations of the rotor in this configuration (estimated in the calculations to be equal to approx. *n = 60Hz* - cf. Figure 26b) does not occur in practice in the results of the experimental investigations. The investigations carried out when the rotor did not move have proven a significant influence of the auxiliary magnetic bearing on rotor dynamics.

Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 189

In Figure 27 cascade plots of the journal vibrations in the magnetic bearing during the startup of the test stand in the so-called "full spectrum" domain are presented. A comparison of the diagrams illustrates the role played by a presence of the auxiliary magnetic bearing from the point of view of shaft vibrations during the start-up (or shut-down) of the model

**Figure 27.** Cascade plots of the journal vibrations in the magnetic bearing during the start-up of the model shaft line a) auxiliary magnetic bearing disactivated, b) auxiliary magnetic bearing activated

vibrations, dangerous for the machine operation, appear.

**8. Conclusions** 

and actuators [11-14].

experimental investigations are discussed.

The presented idea of an application of the active magnetic bearing as a system that modifies the dynamic properties of the rotating system can turn out to be an interesting alternative to modernisation of a real rotary machine. The majority of overcritical failures of rotors is caused by a local increase in the synchronous or asynchronous vibration amplitude and by exceeding the clearances when the critical frequency or the stability threshold of the rotating system is exceeded. The special character of the active magnetic bearing operation makes it possible to activate it when the shaft rotates, whereas relatively high values of the clearances allow for locating it practically in any place of the machine shaft line. The built test stand enabled practical realisation of the idea of "omitting" the region in which rotor

The progress in the field of active magnetic bearings is based on new operating and diagnostic capabilities of these digitally controlled bearings in comparison to classical solutions. The overall goal of an *AMB* controller is to stabilise the plant and to reach the optimal technical performance. To achieve these goals, *AMB* systems have to be optimised in an overall mechatronics design approach. This leads to a new concept for control systems

In this work, an idea of the simulation model of magnetic levitation systems and its diagnostic capabilities is presented. Some results of the numerical simulations and

machine.

They have shown possibilities of obtaining an abrupt change in the dynamic properties of the rotating system, which complies with the idea of the safe exceeding of the critical frequency of the machine.

In the second stage of the experimental investigations, the dynamics of the model rotating system with a synchronous excitation with unbalance forces was analysed. The investigations were carried during the start-up and shut-down of the system rotor. Numerical simulations of the test stand rotor response to unbalance were performed. An example of the calculation results shown in Figure 26 indicates that for the nominal rotational frequency *50Hz*, the test stand rotor can be an overcritical or undercritical system, depending on the fact whether the auxiliary magnetic bearing characterised by specified dynamic properties is activated or disactivated.

**Figure 26.** *A* – Experimental Bode plot *B* – Theoretical Bode plot *a* – start-up, the rotating system supported in ball bearings, *b* – shut-down, with the auxiliary magnetic bearing activated

An application of the auxiliary magnetic bearing allowed for a complete elimination of a dangerous increase (for the whole local system operation) in the vibration amplitude when the critical frequency had been exceeded. The optimal moment for activation and disactivation of the additional bearing system has been indicated by a solid line. At this moment, the vibration amplitudes of both the qualitatively different dynamic systems reach similar values. In the presented example, the switching took place above the optimal values of revolutions, which was connected with an abrupt change in the vibration amplitude and phase.

In Figure 27 cascade plots of the journal vibrations in the magnetic bearing during the startup of the test stand in the so-called "full spectrum" domain are presented. A comparison of the diagrams illustrates the role played by a presence of the auxiliary magnetic bearing from the point of view of shaft vibrations during the start-up (or shut-down) of the model machine.

**Figure 27.** Cascade plots of the journal vibrations in the magnetic bearing during the start-up of the model shaft line a) auxiliary magnetic bearing disactivated, b) auxiliary magnetic bearing activated

The presented idea of an application of the active magnetic bearing as a system that modifies the dynamic properties of the rotating system can turn out to be an interesting alternative to modernisation of a real rotary machine. The majority of overcritical failures of rotors is caused by a local increase in the synchronous or asynchronous vibration amplitude and by exceeding the clearances when the critical frequency or the stability threshold of the rotating system is exceeded. The special character of the active magnetic bearing operation makes it possible to activate it when the shaft rotates, whereas relatively high values of the clearances allow for locating it practically in any place of the machine shaft line. The built test stand enabled practical realisation of the idea of "omitting" the region in which rotor vibrations, dangerous for the machine operation, appear.
