**3. Numerical model of the magnetic bearing system**

166 Performance Evaluation of Bearings

machine operation.

**2. Digitally controlled magnetic bearing** 

Active magnetic suspension systems of machine rotors being built at present are equipped with digital control systems. Apart from a possibility of implementation of complex control algorithms, they provide also wide diagnostic possibilities resulting from an application of measurement techniques at different stages of the system design. Control systems of bearing responses decide about dynamic properties of the rotating system. Digital controllers allow for, e.g. a change in bearing dynamic properties during motion in different modes of the

**Figure 2.** System of the digitally controlled magnetic bearing (one of the control axes - *y*).

The research on active magnetic bearing technology, including works on digital controllers and algorithms, actuators and magnetic bearing-rotor system dynamics, has been carried out for several years in the Institute of Turbomachinery of the Technical University of Łodz [4-9]. The mechanical structure of the built active magnetic bearing, consists of a journal and a bush with four pairs of electromagnets placed equally around the rotor. The position of ferromagnetic journal (1) with respect to bush (2) is controlled by means of eddy-current displacement transducers (3) made by *Bently Nevada Corporation*, with the diameter *d=8 mm*  and the static sensitivity *7.870 V/mm*. They are mounted on two control axes *x, y* that are perpendicular with respect to each other and displaced by the angle 450 with respect to the journal axis. The control axes interact with respective pairs of electromagnets (Figure 2).

Each pair of the bush electromagnets of the journal bearing interacts with a digitally controlled power amplifier with a variable pulse width *PWM* (5)*.* The control pulse-width modulation *Wy* and *Wx* is counted by controller (4) on the basis of measurements of the position *y* and *x* of the journal, respectively. The control current that supplies the windings The basic assumption while developing the numerical model of the bearing system was to offer a tool that allows for tuning the parameters of its controller and for carrying out the investigations of the designed system dynamics in a wide range [4,14]. The condition to be met during the realisation of the idea of the numerical simulation of the bearing was to reproduce the algorithm of operation of the real bearing actuating system structure, and the measurement and control elements applied in this real system in the model. The fulfilment of this requirement has guaranteed the correctness of the operation of the model and feasibility of its design.

A general scheme of the system that has been employed in order to develop the numerical model is presented in Figure 3a. In this model, a motion of the mass *m* concentrated in the geometrical center of the journal is analyzed.

A controller used in the system controls both the axes *x* and *y*. For each axis, differential control with a programmed value of the *pulse-width modulation* of the so-called *WB* base has been applied. Figure 3b shows system of the position of the journal with respect to the bearing bush: *EM1, EM2* – bush electromagnets interacting with the axis *y, EM4, EM3* – bush electromagnets interacting with the axis *x, FZ* – rotating vector of residual unbalancing, *Fm x* – electromagnet force acting along the axis *x, Fm y* – electromagnet force acting along the axis *y, Fg + Fstat –* forces of gravity and static load.

**Figure 3.** a. - Diagram of the model conception, b. - Distribution of forces (one of the control axes - *y*)

An idea of this model assumes a possibility of numerical simulations of dynamic properties of the system, owing to three defined levels of data connected with the bearing system structure, namely:


Theoretical and Experimental Investigations of

Dynamics of the Flexible Rotor with an Additional Active Magnetic Bearing 169

electric circuit versus time (i.e., in subsequent periods of control pulses) for nominal parameters of the power amplifier elements, electromagnet windings, supply voltage and

An operation of the actuator was analyzed. Each cycle of the control pulse *PWM* of a given frequency *fPWM* and a pulse-width modulation *W* forces two modes of the amplifier

charging, whose duration is equal to *tchar = W / fPWM .* Elements of the amplifier circuit

discharging, whose duration is equal to *tdischar = (1* - *W) / fPWM* . Elements of the amplifier

For a given pulse-width modulation, a value of the control current *Iav i* that generates the bearing magnetic force results from its averaged value in the *PWM* cycle and has been

The coefficient *Xi* occurring in formula (1) determines a ratio of the discharging time for a given cycle to the time during which the power amplifier electronic elements cause that the control current value diminishes to zero. In the cycles in which the current discharges to *Imin i* 

Individual operation sequences of the digitally controlled pulse power amplifier and the windings powered by it have to be analyzed. Numerically simulated sequences of the ideal actuator operation enable one to generate model characteristics, which are the basis for evaluation of an influence of changes in values of individual parameters of the actual system

*avi i i i*

*I0 i –* initial current for the *i – th* cycle of the amplifier charging. *I0 i+1 –* initial current for the *i+1 – th* cycle of the amplifier charging.


*I I I I I W W X* (1)

**cykle PWM**

*cycles PWM*

**prad elektromagnesu impuls PWM** *pulse width PWM*

*current*

**Figure 4.** Power amplifier circuit and an idea of current changes in the bearing

control frequency, was required.

operation, namely:

expressed by relation (1):

*= I 0 i+1* , the coefficient *Xi =1.*

on their shape.

operate then in the conduction mode,

circuit operate in the lockout mode.


The model gives also a possibility of analysis of permissible levels of disturbances *FZ, Ru, No* that provide a proper margin of the bearing system stability for a given type of the controller. Thus, in a sense, it constitutes the synthesis of a robust controller.

The numerical procedures representing actual characteristics of the actuator were developed and verified, and then applied in the simulation model of the bearing. This allows for modelling the magnetic bearing system quickly and accurately. The numerical simulation of the active magnetic bearings system by means of the Hewlett-Packard HPVEE software was elaborated.

In the designed digital control system of a journal active magnetic bearing, the suitable software that allows for investigations of the *pulse-width modulation - control current* characteristics that determine the bearing system properties while its actuators are built, has been applied.
