Active Control of an Unbalanced Rotor System using Active Bearings and the Generalized Disturbance Estimation Liaoyuan Ran 1 , Dunant Halim 2 *, Tingyu Lin 3 Department of Mechanical, Materials and Manufacturing Engineering The University of Nottingham Ningbo China Ningbo, Zhejiang, China 315100 Chung Ket Thein 4 School of Aerospace The University of Nottingham Ningbo China Ningbo, Zhejiang, China 315100 Michael Galea 5 Department of Industrial Electrical Power Conversion University of Malta Msida, M.S.D., Malta 2080

ABSTRACT The work is aimed to develop an active control method for suppressing unbalanced vibration of an elastic rotor system via the utilization of active bearings. The active bearing was implemented by the developed control system to actively control the displacements of the bearing for regulating the bearing forces. To address the issue of uncertainties associated with the unbalanced forces and internal dynamics of the rotor system, the Extended State Observer (ESO) was utilized to provide an accurate estimation of the generalized disturbance that consisted of the external disturbance in the form of radial forces caused by the unbalanced rotating mass of the rotor, and the disturbance associated with uncertainties in the internal dynamics of the rotor system. Active Disturbance Rejection Control (ADRC) method was utilized to cancel the effect of the generalized disturbance, allowing a simpler control input implementation to the active bearing. It was shown in this work that the ESO was able to accurately estimate the disturbances affecting the rotating system in real time. As the result, the active bearing control system could reject the disturbances effectively to minimize the unbalanced vibration of the rotor system.

1 liaoyuan.ran@nottingham.edu * 2 dunant.halim@nottingham.edu.cn (Corresponding author) 3 tingyu.lin@nottingham.edu.cn 4 chungket.thein@nottingham.edu.cn 5 michael.d.galea@um.edu.mt

1. INTRODUCTION

In the past decades, vibration suppression of rotor systems has been addressed either by using passive [1], semi-active [2], or active control methods [3]. A rotor system is one of the core components of a power transmission system, which has been utilized in a wide range of industrial applications. Due to the inherent flexibility of a rotor structure, excessive vibration can lead to performance degradation and machine faults particularly when the rotor system operates under high- speed operating conditions [4], whose vibration can be caused by the misalignment of couplings, unbalanced rotor, and cracks in the shaft [5-7]. Therefore, a suitable vibration attenuation strategy should be applied to the rotor system for its high-speed operation.

Chen et al. [8] proposed an active vibration control strategy that used three magnetostrictive actuators directly mounted on the rotor shaft to control the steady-state torsional vibration. It has been shown from experiment that 7 dB noise reduction could be achieved, which was associated with gear vibration at the basic tooth mesh frequency of 250 Hz. In [9], a PID controller was developed for vibration suppression of a flexible rotor system supported by two active magnetic bearings (AMB). Experimental results showed that the developed control system had sufficient stability robustness for the attenuating vibration close to the resonance. Comparative studies on control performances using the polar fuzzy, fuzzy PID, and PID controllers on a flexible rotor supported by two AMBs running above three critical speeds, were presented in [10]. It was concluded that the polar fuzzy control has a better performance under an identical rotor system configuration. However, the application of AMB generally requires a relatively complex control system with an additional internal feedback control to stabilize the levitation of the shaft. Alternatively, active bearings have shown their potential for vibration control of rotor systems since 1984, utilizing piezoelectric-type of actuators in connection with conventional springs and bearings [11]. Despite the active bearing control whose potential for vibration control has been demonstrated to simple rotor systems, the challenge remains on how to ensure the control system has the sufficient robustness in suppressing vibration in a rotor system, whose accurate model may not be known.

Active disturbance rejection control (ADRC) was initially proposed in 1998 [12], which has been demonstrated to be relatively robust against uncertainties in the internal dynamics of the plant and external disturbances. By using an Extended State Observer (ESO), the generalized disturbance affecting the system can be estimated and compensated [13]. ADRC has shown its tremendous potential for a wide range of applications, such as for motion control of robotic systems, speed control of electrical machines, and vehicle suspension control. In this work, therefore, active control of an unbalanced rotor system using active bearings and ADRC approach is proposed for effective vibration suppression.

2. SYSTEM MODELING

The Jeffcott rotor system can be used to describe a common rotor system with unbalanced mass, as shown in Figure 1. Two active bearings are attached to both ends of the shaft to constrain the shaft motion in the radial direction, and a driving motor is connected to a rotor system with a flexible coupling. The active bearing consists of two pre-load springs and two piezoelectric actuators, which are linked in series to provide the support stiffness of the rotor system in x and y directions. The stiffness of the pre-load spring and the actuator form the overall actuator stiffness 𝑘 𝑎 . The equivalent stiffness and internal damping of the flexible shaft are represented by 𝑘 and 𝑐 . For a homogenous shaft, the stiffness and damping are assumed to be the same in x and y directions. The rigid disc located in the shaft center has a mass eccentricity. The mass of the shaft is assumed to be much smaller compared to the mass of the rigid disc, so it can be ignored in this case.

Figure 1: An unbalanced Jeffcott rotor system with active bearings. The displacement 𝑋 1 generated from the piezoelectric actuator has a linear relationship with the control voltage 𝑉 𝑎 . The control force produced by the active bearing is expressed by 𝐹 𝑎𝑥 and 𝐹 𝑎𝑦 , where the subscripts denote the control force in the respective x or y direction. As the rotor system rotates with an angular speed of 𝜔 , the unbalanced force due to the unbalanced mass can be represented by:

𝐹 𝑢 = 𝑚𝑒𝑤 2 (1) where m is the mass of the rigid disc, and 𝑒 is the mass eccentricity. In Figure 2, a simplified diagram of the coupled rotor system with the active bearing is depicted. The equation of motion of the rotor system with active bearings can be described in four degrees of freedom associated with the vertical and horizontal displacements expressed by 𝑋 , 𝑌, 𝑋 1 , and 𝑌 1 .

Active bearing Active bearing 2 Driving motor AT*NMET Uabatanced rotor id LJ jl Flezible coupling

Figure 2: The cross-sectional view of a simplified rotor system with active bearings.

The gyroscopic effect is ignored in this case as the disc is located at the center of the shaft. Then, the equation of motion of the rotor system can be described as:

𝑚𝑋̈ + 𝑐(𝑋̇ −𝑋 1 ̇ ) + 𝑘(𝑋−𝑋 1 ) = 𝐹 𝑢 𝑐𝑜𝑠𝜔𝑡

𝑚𝑌 ̈ + 𝑐(𝑌 ̇ −𝑌 1 ̇ ) + 𝑘(𝑌−𝑌 1 ) = 𝐹 𝑢 𝑠𝑖𝑛𝜔𝑡

(2)

−𝑐(𝑋 ̇ −𝑋 1 ̇ ) −𝑘(𝑋−𝑋 1 ) + 𝑘 𝑎 𝑋 1 = 𝐹 𝑎𝑥

{

−𝑐(𝑌 ̇ −𝑌 1 ̇ ) −𝑘(𝑌−𝑌 1 ) + 𝑘 𝑎 𝑌 1 = 𝐹 𝑎𝑦

The transfer functions from 𝐹 𝑎𝑥 to 𝑋, and 𝐹 𝑎𝑦 to 𝑌 can be expressed by:

𝑋(𝑠)

𝑐𝑠+𝑘

𝑚𝑐𝑠 3 +(𝑚𝑘+𝑚𝑘 𝑎 )𝑠 2 +𝑐𝑘 𝑎 𝑠+𝑘𝑘 𝑎 (3)

𝐹 𝑎𝑥 (𝑠) =

(pis

𝑌(𝑠)

𝑐𝑠+𝑘

𝐹 𝑎𝑦 (𝑠) =

(4)

𝑚𝑐𝑠 3 +(𝑚𝑘+𝑚𝑘 𝑎 )𝑠 2 +𝑐𝑘 𝑎 𝑠+𝑘𝑘 𝑎

In the present work, the active bearings located at both ends of the shaft will generate similar control actuation. To regulate the rotational motion of the rotor system, the active bearings generate controlled displacements of the bearing to eliminate the bearing force produced by the unbalanced mass. 3. POSITION CONTROL WITH ROTOR POSITION FEEDBACK

The main working principle of ADRC is that it uses the ESO to estimate the unknown generalized disturbance, consisting of internal and external disturbances, which will be cancelled out to allow a simpler control law implementation. To control the lateral displacements of the rotor system, Eqs. (3) and (4) can be rewritten as

𝑋 ⃛ + 𝑎 1 𝑋 ̈ + 𝑎 2 𝑋 ̇ = 𝑏 1 𝑢̇ + ℎ (6) where 𝑋 is the displacement of the rotor system; 𝑢 is the control input applied to the system; ℎ is the generalized disturbance; and 𝑎 1 , 𝑎 2 , and 𝑏 1 can be expressed as:

( 𝑚𝑘+𝑚𝑘 𝑎 )

𝑘 𝑎

1

𝑚 (7) A second-order system can be obtained by integrating both sides of Eq. (6) and re-written it as:

𝑎 1 =

𝑐𝑚 , 𝑎 2 =

𝑚 , 𝑏 1 =

𝑋 ̈ = 𝑏 1 𝑢+ 𝑓(𝑋 ̇ , 𝑋, ∫ℎ) (8) By defining the states 𝑥 1 = 𝑋 , 𝑥 2 = 𝑋 ̇ , 𝑥 3 = 𝑓 , a standard state-space representation of Eq. (8) is given as:

{ 𝑥̇ = 𝐴𝑥+ 𝑏 1 𝐵𝑢+ 𝐸𝑓̇

𝑋= 𝐶 𝑐 𝑥+ 𝐷𝑢 (9)

0 1 0 0 0 1 0 0 0

0 1 0

0 0 1

where 𝐴= [

] , 𝐶 𝑐 = [1 0 0], 𝐷= [0] .

] , 𝐵= [

] , 𝐸= [

The third-order ESO can then be designed as:

𝑧̇ = 𝐴𝑧+ 𝑏 1 ̂ 𝐵𝑢+ 𝐿(𝑥 1 −𝑧 1 ) (10) where 𝑏 1 ̂ is the estimated value of 𝑏 1 ; 𝐿= [𝐿 1 𝐿 2 𝐿 3 ] is the gain vector of the observer; 𝑥 1 −𝑧 1 is the observer error from the ESO. It is suggested in [14] that the observer gain has the following relationship to the bandwidth 𝜔 𝑜 of the controller:

𝐿 1 = 3𝜔 𝑜 𝐿 2 = 3𝜔 𝑜

2

(11)

{

3

𝐿 3 = 𝜔 𝑜

A Proportional-Derivative (PD) control law can then be implemented as:

𝑢 0 −𝑧 3

𝑏 ̂ 1 (12)

𝑢(𝑡) =

𝑢 0 = 𝑘 𝑝 (𝑥−𝑧) + 𝑘 𝑑 (𝑥̇ −𝑧̇) (13) where 𝑘 𝑝 is the proportional controller gain, while 𝑘 𝑑 is the derivative controller gain. The values of 𝑘 𝑝 and 𝑘 𝑑 are set as:

2

{ 𝑘 𝑝 = 𝜔 𝑐

(14)

𝑘 𝑑 = 2𝜔 𝑐

where 𝜔 𝑐 is the bandwidth of the controller. By substituting Eq. (12) into Eq. (8), the generalized disturbance 𝑓 can be compensated by the estimated 𝑧 3 , so the rotor system can be directly controlled by using a PD controller. 4. PERFORMANCE EVALUATION OF THE ACTIVE BEARING CONTROL SYSTEM

4.1 The open-loop rotor system

By assigning the rotor system with specific values: 𝑚= 1 Kg, 𝑐= 0.5 Ns/m, 𝑘= 100 N/m, 𝑘 𝑎 = 1000 N/m, and 𝑒= 0.01 , the estimated natural frequency from Eq. (5) is calculated as 9.53 rad/s, as observed in the frequency response of the open-loop rotor system in Figure 3.

Figure 3: Frequency response of the open-loop rotor system. 4.2 ESO accuracy in the generalized disturbance estimation

To investigate the performance of the observer, an impulse force with a magnitude of 10 N is applied to the rotor system at 0.5 s, representing an external disturbance. Figure 4(a) shows the time response of the generalized disturbance (System state 3) and its estimation (Observer state 3). The estimation error is plotted in Figure 4(b). It is observed that the error initially increases to 3.79e10 -8 at 0.5s when the disturbance is applied to the system, but then it significantly decreases. The estimated state 𝑧 3 accurately tracks the state 𝑥 3 , which indicates a satisfactory performance of the developed observer.

7 i i xl0* (w) aponydany 0203 04 05 06 07 Os 09 Time(s) or

(a) (b) Figure 4: (a) Time responses of the system state 𝑥 3 (the generalized disturbance) and the observer

state 𝑧 3 (the estimated generalized disturbance). (b) Time response of the estimation error.

4.3 Effects of the controller and observer bandwidths on control performance

In this section, the effects of controller and observer bandwidths on active bearing control performance are evaluated. The active bearing control system with different controller and observer bandwidths is compared under an excitation frequency of 𝜔 =50 rad/s with an eccentricity 𝑒 =0.01. Another disturbance is applied to the system with an impulse force of 10N at 0.1 s. Figures 5 (a) and (b) show the rotor displacement under the two excitation forces.

10% os 020288 04 Time(s)

(a) (b) Figure 5: Rotor responses for the rotor system with active bearing control:

(a) Tracking performance. (b) Disturbance rejection. The control performances of three different sets of controller and observer bandwidths are shown in Table 1. From Figure 5, it can be seen that a low bandwidth setting of the controller and observer can bring in a relatively poor control performance, which shows the importance of selecting suitably high bandwidths according to the operational bandwidth of the rotor system.

Amplitude (m) oa 042 on 046 048 05 Time(s) 032 ost 056 088 06

Table 1. Control performances for different control and observer bandwidths. 𝜔 𝑜 (rad/s)

Max Error (%) 5 ‰ Settling Time (ms) Overshoot ( ‰ )

𝜔 𝑐 (rad/s)

500 1000 40.3% 2.13 0.036 1000 2000 6.27% 1.37 0.005 1500 3000 1.72% 0.76 0.002

x1o% as 2s 1s os oa 042 044 046 O48 0s Time (5) 052 0st 056 058 06

4.4 Active bearing control performance with different rotor eccentricities and masses

Since the eccentricity of the rotor system is often not accurately known, it can be treated as part of the external disturbance to be rejected by the developed active bearing control system. Unknown variation of the rotor mass can be treated as uncertainty in the rotor system's internal dynamics, considered an internal disturbance. In this section, a rotor system with different eccentricities and masses is investigated to evaluate the robustness of the active bearing control system. The bandwidths of the controller and observer are selected from the previous section: 𝜔 𝑜 = 1500 rad/s and 𝜔 𝑐 = 3000 rad/s. When the resonance of the rotor system at frequency of 9.53 rad/s is excited, the rotor displacement is quite large when there is no control applied. Therefore, to evaluate the performance of the active bearing control, the rotor displacements under different rotor eccentricities and masses are compared. For the first scenario, the rotor system with different mass eccentricities e (0.01, 0.1, 1) is excited at its resonance. For the second scenario, the rotor system with different rotor masses 𝑚 (1 kg, 2 kg, 4 kg) and the eccentricity e of 0.01, is used instead. Figures 6(a) and (b) show the rotor responses for the first and second scenarios, respectively, while Table 2 indicates the maximum rotor displacement for different mass eccentricities.

Amplitude (m) os 0s pal 2 23 Time (s) 3 35

(a) (b) Figure 6: Rotor responses under: (a) Different rotor eccentricities. (b) Different rotor masses.

Table 2. Rotor displacement with/without active bearing control

Maximum rotor displacement with

Maximum rotor

Eccentricity

Mass (Kg)

displacement with control (m) 0.01 8.80 × 10 −8 1 8.792 × 10 −8 0.1 8.80 × 10 −7 2 1.761 × 10 −7 1 8.80 × 10 −6 4 3.521 × 10 −7 It can be seen from Table 2 that the maximum rotor displacement under active bearing control is within an acceptable range. Although an increase of rotor mass and eccentricity can bring in an increase of maximum rotor displacement, the maximum rotor displacement is still relatively small, below 0.01mm. The results demonstrate the effectiveness of active bearing control in suppressing vibration of a rotor system at its resonance, even under the variation of rotor mass and eccentricity. 5. CONCLUSIONS

control (m)

In this study, ADRC-based active bearings have been developed to suppress vibration in a rotor system. The active bearing has the potential to be used for active vibration control of rotor systems

Amplitude (m) x10"

due to its relatively simple control structure compared to AMB, with its capability for effective bearing displacement control to eliminate the bearing forces associated with the unbalanced force. By considering the active bearing control of a Jeffcott rotor system, the developed ESO has been shown to be able to accurately estimate the generalized disturbances that are contributed by both internal and external disturbances affecting the rotor system. The simulation results have also demonstrated the robustness of active bearing control utilizing ADRC, to provide effective vibration suppression of a rotor system that is affected by internal and external disturbances. 6. ACKNOWLEDGEMENTS

This work was supported by Ningbo Science and Technology Bureau under Natural Science Programme (Project code 202003N4183) China. 7. REFERENCES

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