**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

electromagnets and alters in each control cycle *PWM* [5,9]. The value of the magnetic response component *FXmag* for one control axis is related to the measured mean values of the current controlling the electromagnets *IXT, IXB* in a given control period and the values of the magnetic gaps *sXT, sXB* (top – index *T*, bottom – index *B*). The values of the magnetic gaps are found on the basis of measurements of instantaneous positions of the journal with respect to the centre of the bush of the known clearance. The magnetic response component *FXmag* for the axis *X* is determined by the following relationship:

$$F\_{X\text{ }mag} = K\_{XT} \frac{{I\_{XT}}^2}{{s\_{XT}}^2} - K\_{XB} \frac{{I\_{XB}}^2}{{s\_{XB}}^2} \tag{2}$$

Theoretical and Experimental Investigations of

X-Y

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Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 179

the bearing response force, is employed on the assumption of a lack of coupling between the

Fx lin = KXX xac + CXX vx (3)

A difference between the non-linear magnetic bearing response force in a given axis *Fxmag ac*, known on the basis of model calculations and its linear form *Fx ac lin* determined by the

Fx lin - Fxmag ac = F (4)

Changes in the non-linear magnetic response force *Fmx ac*, provided by the digital controller for a given control axis, which result from the numerical calculations, are approximated with the linearized harmonic time history *Fx ac lin*. Its values are determined by the dynamic stiffness *KXX* and damping coefficients *CXX* for a given control axis. An analogous situation

It is possible to use the proposed method for calculation of dynamic coefficients of the bearing when the developed simulation model of the bearing, whose operation is

The calculations are conducted for stable bearing operation, where the journal position oscillates around the assumed point of equilibrium and the interactions between the control

[ s ]

Figure 15 presents a comparison between the measured magnetic response force *FXmag* and the theoretical function, which is a sum of the forces of stiffness and damping *FX lin = KXX x +* 

The curve *FX lin* has been plotted on the basis of the measured journal displacement *x* (Figure 14) and the journal velocity *VX* obtained through digital differentiation of the displacement and a selection of suitable values of the dynamic stiffness coefficients *KXX* and the damping

axes *X* and *Y* can be neglected for small displacements of the journal (Figure 14).

coefficients of stiffness and damping *KXX [N/m], CXX [Ns/m]* are obtained.

convergent with the operation of a real bearing system, is employed.

*F2 = min* is sought with the least squares method. Thus, the linearized

bearing control axes. For one control axis:

refers to the coefficients *KYY , CYY* for the axis *Y.*

**Figure 14.** Displacements *X, Y* versus time and the orbit

0.05 0.10 0.15 0.20 0.25

[ um ] X Y

formula:

*CXX VX*.


in such a way that

Equation (2) holds on the assumption that the linear dependence of the magnetic flux on the induction is maintained. It means that the bearing operates according to this part of the characteristics that is distant enough from the state of magnetic circuit saturation, when the induction does not exceed 50 % of the saturation induction for the core material. The value of the constant *K* depends on electromagnet design parameters and can be calculated theoretically [1]. However, in the actual design of the journal bearing, the constants *KXT, KXB KYT, KYB* can differ slightly for each pair of electromagnets of the bush. In order to increase the accuracy of the proposed measurement method, the constant values are verified experimentally for each electromagnet and their real values are taken into account in the calculations [8]. If the journal motion parameters are known and the magnetic response force is determined by an indirect method, it is possible to find the bearing dynamic parameters that relate the magnetic response force to the journal motion parameters [5,9].

An analysis of the system response to synchronous excitation points out to the fact that the bearing magnetic response force *Fxmag Fymag* is proportional to the excitation force amplitude *Fz* in the whole range of frequency of rotations. It allows for the identification of equivalent dynamic coefficients of the bearing. In the proposed method a non-linear analysis of the simulation model in the assumed range of rotational frequency changes of the rotating mass of the bearing, for the excitation with the rotating vector of unbalance *FZ,* is required. For each assumed value of the frequency of rotations, the value of unbalance *FZ* is chosen in a such a way that the forced vibrations of the bearing rotating mass have the same value of amplitude in each control axes *x, y.* For the assumed frequency values *n*, the time histories of the following quantities are recorded:


These time histories are non-linear, as no simplified assumptions have been introduced into the model for the relations between the force, current and displacement. In the final stage of the proposed method, a relation between the linear stiffness and damping coefficients and the bearing response force, is employed on the assumption of a lack of coupling between the bearing control axes. For one control axis:

178 Performance Evaluation of Bearings

the axis *X* is determined by the following relationship:

the following quantities are recorded: - excitation unbalance force *FZ (t)*,

*xac(t), yac(t)*,

*ac(t)*.

electromagnets and alters in each control cycle *PWM* [5,9]. The value of the magnetic response component *FXmag* for one control axis is related to the measured mean values of the current controlling the electromagnets *IXT, IXB* in a given control period and the values of the magnetic gaps *sXT, sXB* (top – index *T*, bottom – index *B*). The values of the magnetic gaps are found on the basis of measurements of instantaneous positions of the journal with respect to the centre of the bush of the known clearance. The magnetic response component *FXmag* for

*X mag XT XB*

Equation (2) holds on the assumption that the linear dependence of the magnetic flux on the induction is maintained. It means that the bearing operates according to this part of the characteristics that is distant enough from the state of magnetic circuit saturation, when the induction does not exceed 50 % of the saturation induction for the core material. The value of the constant *K* depends on electromagnet design parameters and can be calculated theoretically [1]. However, in the actual design of the journal bearing, the constants *KXT, KXB KYT, KYB* can differ slightly for each pair of electromagnets of the bush. In order to increase the accuracy of the proposed measurement method, the constant values are verified experimentally for each electromagnet and their real values are taken into account in the calculations [8]. If the journal motion parameters are known and the magnetic response force is determined by an indirect method, it is possible to find the bearing dynamic parameters that relate the magnetic response force to the journal motion parameters [5,9].

An analysis of the system response to synchronous excitation points out to the fact that the bearing magnetic response force *Fxmag Fymag* is proportional to the excitation force amplitude *Fz* in the whole range of frequency of rotations. It allows for the identification of equivalent dynamic coefficients of the bearing. In the proposed method a non-linear analysis of the simulation model in the assumed range of rotational frequency changes of the rotating mass of the bearing, for the excitation with the rotating vector of unbalance *FZ,* is required. For each assumed value of the frequency of rotations, the value of unbalance *FZ* is chosen in a such a way that the forced vibrations of the bearing rotating mass have the same value of amplitude in each control axes *x, y.* For the assumed frequency values *n*, the time histories of



These time histories are non-linear, as no simplified assumptions have been introduced into the model for the relations between the force, current and displacement. In the final stage of the proposed method, a relation between the linear stiffness and damping coefficients and


*I I FK K*

2 2 2 2 *XT XB*

*s s* (2)

*XT XB*

$$\mathbf{F}\_{\text{x}}\mathbf{\lim} = \mathbf{K}\boldsymbol{\chi}\boldsymbol{\chi}\mathbf{x}\_{\text{ac}} + \mathbf{C}\boldsymbol{\chi}\boldsymbol{\chi}\mathbf{v}\_{\text{x}} \tag{3}$$

A difference between the non-linear magnetic bearing response force in a given axis *Fxmag ac*, known on the basis of model calculations and its linear form *Fx ac lin* determined by the formula:

$$\mathbf{F}\_{\text{x-lim}} - \mathbf{F}\_{\text{x-mag}} \mathbf{a} = \Delta \mathbf{F} \tag{4}$$

in such a way that *F2 = min* is sought with the least squares method. Thus, the linearized coefficients of stiffness and damping *KXX [N/m], CXX [Ns/m]* are obtained.

Changes in the non-linear magnetic response force *Fmx ac*, provided by the digital controller for a given control axis, which result from the numerical calculations, are approximated with the linearized harmonic time history *Fx ac lin*. Its values are determined by the dynamic stiffness *KXX* and damping coefficients *CXX* for a given control axis. An analogous situation refers to the coefficients *KYY , CYY* for the axis *Y.*

It is possible to use the proposed method for calculation of dynamic coefficients of the bearing when the developed simulation model of the bearing, whose operation is convergent with the operation of a real bearing system, is employed.

The calculations are conducted for stable bearing operation, where the journal position oscillates around the assumed point of equilibrium and the interactions between the control axes *X* and *Y* can be neglected for small displacements of the journal (Figure 14).

**Figure 14.** Displacements *X, Y* versus time and the orbit

Figure 15 presents a comparison between the measured magnetic response force *FXmag* and the theoretical function, which is a sum of the forces of stiffness and damping *FX lin = KXX x + CXX VX*.

The curve *FX lin* has been plotted on the basis of the measured journal displacement *x* (Figure 14) and the journal velocity *VX* obtained through digital differentiation of the displacement and a selection of suitable values of the dynamic stiffness coefficients *KXX* and the damping

coefficients *CXX* in such a way as to make the sum of squares of differences minimal for the selected part of the time history.

Theoretical and Experimental Investigations of





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20

60

100

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Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 181

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

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

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

**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

**[ s ]**

**[ um ] Xac Yac orbita X-Y**

maximum sampling frequency of *100kHz*, was applied.

signals *0-5V* from current-voltage *LEM* type transducers.

0.20 0.22 0.24 0.26 0.28 0.30

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

response forces.




20

60

100

**Figure 15.** Measured magnetic response component along the control axis *X* - *FXmag* and its modelled time history *FX lin* with the identified dynamic coefficients *KXX , CXX*

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