Robust modal vibration suppression of a flexible rotor

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Mechanical Systemsand

Signal Processing

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Mechanical Systems and Signal Processing 21 (2007) 334–347

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Robust modal vibration suppression of a flexible rotor

Hsiang-Chieh Yua, Yih-Hwang Linb,, Chih-Liang Chuc

aIndustrial Technology Research Institute, Opto-Electronics & Systems Laboratories, Chutung, Hsinchu 310, Taiwan, Republic of ChinabDepartment of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 20224, Republic of China

cDepartment of Mechanical Engineering, Southern Taiwan University of Technology, Tainan 710, Taiwan, Republic of China

Received 13 April 2005; received in revised form 14 October 2005; accepted 18 October 2005

Available online 29 November 2005

Abstract

This study deals with active robust modal vibration control of rotor systems supported by magnetic bearings. The

inherent divergent rigid body modes are suppressed by using a dual-level control approach. Finite element method is

applied to formulate the rotor model. The Timoshenko beam theory, including the effects of shearing deformations and

rotary inertia, is considered in this work. Because practical control systems are often limited by its sensing, hardware, and

computation speed capabilities, the reduced order approach is often used for a control system design. This study applies

the independent modal space control (IMSC) approach to extract the accurate lower modes from the complex rotor

systems with the gyroscopic effect considered. In practice, it is extremely difficult to model the complete dynamic

characteristics of a rotor system. The model may contain unmodelled dynamics and parameter changes, which can be

viewed as uncertainties of a system. As opposed to the conventional control approach, which requires fixed and accurate

system parameters, this study considers robust control approach to design a controller capable of tolerating external

disturbance and model uncertainties. It is demonstrated that the proposed approach is effective for vibration suppression

when the system is subjected to impulsive or step loading, speed variation, and sudden loss of disk mass.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Vibration suppression; Robust control; Flexible rotor; Gyroscopic effect

1. Introduction

Magnetic bearings have been shown to possess a number of advantages over the traditional bearings, suchas no need for oil supply systems, no mechanical contact and wear, low power consumption, controllablebearing dynamic properties, excellent positioning accuracy, and long service life, etc. Pole placementtechniques were reported [1] for synchronous vibration control of a rotor-bearing system. An optimisationapproach was presented to minimise the influence of forcing disturbances, modelling error, and measurementerror [2]. Experimental study was performed for unbalance control of a rotor system using adaptive controltechniques [3]. A detailed description of the electromagnetic actuator for active vibration control of a flexiblerotor was presented to reduce the synchronous vibrations of a rotor system [4]. Keogh et al. [5] utilised the

e front matter r 2005 Elsevier Ltd. All rights reserved.

ssp.2005.10.007

ing author. Tel.: +886 2 2462 2192x3208; fax: +886 2 2462 0836.

ess: [email protected] (Y.-H. Lin).

www.elsevier.com/locate/jnlabr/ymssp

ARTICLE IN PRESSH.-C. Yu et al. / Mechanical Systems and Signal Processing 21 (2007) 334–347 335

sampled harmonics for transient vibration control of rotors to alleviate the problem of repeated contacts withthe safety bush. Cole et al. [6] presented the control of multiple frequency components using frequency-matched control signals. The regulator design involving model uncertainty and general external signals usingthe H1 control was reported by Zames [7]. The H1 control approach has been shown to be effective formodels with parameter uncertainties [8,9]. Nonami et al. [10] investigated the vibration control using the H1control approach based on a reduced order model. Cole et al. [11] reported a multivariable controller designwith H1 optimised disturbance rejection criteria for improved tolerance to specific external faults. Forinternal faults, reconfiguration of the control strategy on occurrence of a fault was presented.

In this work, we shall present a dual-level approach for robust independent modal space control (IMSC) of a rotorsystem with gyroscopic effects considered. Notable monographs have been published on the subject of modal control[12,13]. Meirovitch et al. [14] pioneered in the development of IMSC of structures. Lin and Chu [15] proposed a newdesign approach to avoid the severe control spillover problems as found in the works of the IMSC pioneerdevelopers for systems possessing complex modes. This study extends the concept of the dual-level control approach[16]. The original rotor system is transformed to an augmented one in the first level control so that the rigid bodymodes with repeated roots are removed. In the second level control, an H1 controller is designed in the complexmodal space of the augmented system. The analysis cases of the rotor system subjected to impulsive or step loading,speed variation, and sudden mass loss will be examined to explore the applicability of the proposed approach.

The present work synthesises the IMSC and the H1 control for a complex rotor system with semi-definitecharacteristics using a unique dual-level control approach. To the authors’ knowledge, such an endeavor oftechnology fusion has not been reported in the literature. Real application issues will be addressed in thiswork. It is feasible to overcome the inherent non-linear bearing dynamics so that a linear relationship betweenthe input control voltage and the electromagnetic control force can be established. The first level control canbe designed in such a way that the usable frequency range of the actuators is not exceeded by properlychoosing the feedback gains. The use of a modal observer is considered in this work, which alleviates the needfor sensor filtering. The effect of out of band modes is included for analysis by examining the response of thefull system, rather than the reduced system, subjected to the synthesised control input. The controlled modescan never destabilise the out of band modes due to control spillover with the use of the present IMSCapproach. Elaboration of all the above application issues will follow in the subsequent sections.

2. Model development

Fig. 1 shows the rotor system considered in this work. The Timoshenko rotor finite element and the shapefunctions used are shown in Appendix A for completeness. Displacement sensors are located at the bearings

Fig. 1. The rotor system model.

ARTICLE IN PRESSH.-C. Yu et al. / Mechanical Systems and Signal Processing 21 (2007) 334–347336

positions. The actuators are collocated with the sensors. The governing equations of motion can be written as

½Mf €dg O½Gf _dg þ ½K fdg ¼ ½BuðtÞ þ ½EwðtÞ,

fY g ¼ ½Cfdg, ð1Þ

in which

½E ¼ f 0 0 : . . . . . . 1 1 . . . : . . . 0 0 gT4ðNeþ1Þ1, (2)

½B ¼

1 0 . . . . . . . . . 0 0 0 0

0 1 . . . . . . . . . 0 0 0 0

0 0 . . . . . . . . . 1 0 0 0

0 0 . . . . . . . . . 0 1 0 0

26664

37775

T

4ðNeþ1Þ4

, (3)

½C ¼

1 0 0 . . . . . . . . . . . . . . . . . . 0

0 1 0 . . . . . . . . . . . . . . . . . . 0

0 0 0 . . . . . . . . . 1 0 0 0

0 0 0 . . . . . . . . . 0 1 0 0

26664

3777544ðNeþ1Þ

, (4)

where [M], [G], [K], and fdg are the structural mass, gyroscopic damping, stiffness matrices, anddisplacement vector, respectively; ½B and ½E are the coefficient matrices for the control input, u(t),and external disturbances, w(t), respectively; [C] is the distribution coefficient matrix for the displacementsensors.

The material damping of the rotor system, which is not shown in Eq. (1), is assumed small and is neglectedin this analysis. To facilitate the control system design, Eq. (1) is recast in the following state space form:

_xf ðtÞ ¼ Af xf ðtÞ þ Bf uðtÞ þ Ef wðtÞ,

Y f ðtÞ ¼ Cf xf ðtÞ, (5)

where

Af ¼½0 ½I

½M1½K O½M1½G

!; Bf ¼

½0

½M1½B

!; Ef ¼

½0

½M1½E

!,

Cf ¼ ½C ½0

. (6)

In this work, the output feedback control is applied as the first level control to eliminate the repeated rigidbody modes of the original rotor system. Robust IMSC is then applied in the second level to enableindependent control of the targeted complex modes to be controlled. In the process, the 2n coupled first-orderequations are decoupled into n pairs of 2 2 first-order matrix equations with each pair being independent toeach other. The decoupled equations in the modal space can be written as

_qf ¼ Lf qf þQf uðtÞ þ Vf wðtÞ,

Y f ¼ Cf Rqf , (7)

where

Qf ¼ LT Bf , (8)

Vf ¼ LT Ef , (9)

in which L and R represent the left and right modal matrices which consist of the real and imaginaryparts of the left and right eigenvectors, respectively; Lf comprises the real and imaginary parts ofthe eigenvalues of the corresponding complex mode [17]. The first four modes are considered as the target


to be controlled, thus

_qr ¼ Lrqr þQruðtÞ þ VrwðtÞ,

Y r ¼ Crqr, (10)

where

Lr ¼ block-diagonalss os

os ss

" #; s ¼ 1; 2; 3; 4, (11)

Qr 2 R84; V r 2 R84; Cr 2 R48. (12)

For a robust control system as illustrated in Fig. 2, PðsÞ; KðsÞ; w; and u denote the nominal plant, thecontroller, the exogenous input, and the control input, respectively; z ¼ ½Z1 Z2

T represents the controlledoutput; y is the output of the nominal plant; W uðsÞ and W pðsÞ are the weighting function matrices. For systemswith multiplicative uncertainty, the actual control target ~PðsÞ can be represented by the nominal plant PðsÞ andthe multiplicative error Dm as shown below

~PðsÞ ¼ ðI þ DmÞP: (13)

It is a mixed sensitivity problem for designing the present controller. Robust stability and good performancecan be achieved if the following cost function is attained [18–20]

W uðsÞTðsÞ

W pðsÞMðsÞ

1

o1; (14)

where T(s) is the transfer function between w and u, also known as the complementary sensitivity function andM(s) is the transfer function between the disturbance input w and the plant output y. To obtain the H1controller in this work, two simultaneous Riccati equations are solved and the g-iteration search method isapplied for satisfying the inequality conditions during the search process.

In real applications, the effects of actuator dynamics and sensing issues must be considered. In the presentwork, a linear relationship is assumed between the input control voltage and the electromagnetic control force.To realise this condition in practice, the inherent non-linearity of the electromagnetic forces can be linearisedby using an analogue square root circuit [4] for rotors with movement within the vicinity of the equilibrium

Fig. 2. Block diagram of the robust control system.

ARTICLE IN PRESSH.-C. Yu et al. / Mechanical Systems and Signal Processing 21 (2007) 334–347338

point. For the uncontrolled rotor system considered here, the natural frequencies of first four modes areidentically zero. They are altered by the first level control. The feedback gains can be designed in such a waythat the useable frequency range of the actuators is not exceeded.

Modal control requires the feedback of modal states, which can be obtained from the use of a modalobserver [21]. It has been shown that no observation spillover instability due to the modelled residual modesexists and consequently, filtering of the sensor signal is not required. In contrast, in an earlier approach asdiscussed in [22] the observation spillover can be so severe that instability may even occur. Filtering of sensorsoutputs is thus suggested to mitigate the problem.

3. Numerical results

The rotor parameters used for this simulation study is tabulated in Table 1. The operation speed of the rotorruns from 0 to 1000 rad/s (9550 rpm). The robust controller is designed based on a nominal speed at 419 rad/s(4000 rpm). The first level controller utilises the following parameters for the direct output control:

K1;1 ¼ 2:3 10p; K2;2 ¼ 1:3 10p; K9;9 ¼ 1:7 10p; K10;10 ¼ 2:7 10p,

in which p ¼ 4 is used. The resulting natural frequencies of the first four modes are 5.532, 7.141, 12.885, and20.875Hz, respectively. The robust controller serves to provide the second level control. To ensure the robuststability, the weighting function W uðsÞ must be chosen such that the multiplicative error Dm is covered at allfrequency points concerned. Furthermore, gains of the weighting function W pðsÞ are increased in the vicinityof the first four modes for better vibration suppression performance at the lower frequencies. The frequencyresponses of the multiplicative error, Dm, and the weighting functions, W uðsÞ and W pðsÞ are illustrated inFig. 3, with W uðsÞ and W pðsÞ chosen as

W uðsÞ ¼25ðs2 þ 36sþ 900Þ

s2 þ 15sþ 562500I44,

W pðsÞ ¼222:5ðs2 þ 165sþ 4550Þ

s2 þ 45sþ 2025I44. ð15Þ

The controller designed based on the weighting functions chosen must satisfy the condition shown inEq. (14) to ensure robust stability and performance of the closed loop system. The condition is satisfied for thepresent design with the maximum singular value being below unity in the entire frequency range of interest.

Performance of the robust controller is examined by considering the following three different operatingconditions with the same controller design for a nominal system:

Case A: Fig. 4 shows the impulse responses of the rotor system with the first level control only and withrobust control as the second level control when the system is subjected to a unit impulse at the mid-span, i.e. atDOF 5 and DOF 6. The corresponding control input can be seen in Fig. 5. In Figs. 6 and 7 are shown the stepresponses and control input, respectively, when the rotor system is excited by a unit step disturbance withduration of 0.2 s at the mid-span.

Case B: The effect of parameter variations on the performance of the robust control system is examined inthis case. In Figs. 8 and 9 are shown the impulse and step responses, respectively, when the disk mass isdoubled, being unexpected in the initial design of the control system. Figs. 10 and 11 illustrate the impulse and

Table 1

The rotor parameters

Young’s modulus 2 1011N/m2

Mass density 7750 kg/m3

Poisson ratio 0.3

Total rotor length 1.062m

Mass of the disk 13.47 kg

Id of the disk 0.051 kgm2

Ip of the disk 0.102 kgm2

Nominal rotor speed 4000 rpm

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10-1 100 101 102 103-60

-40

-20

0

20

40

60

Max

imum

sin

gula

r va

lue

(dB

)

(rad/s)

Fig. 3. Frequency responses of the multiplicative error and weighting functions. —: DmðsÞ, - - -: W uðsÞ, -.-:W pðsÞ.

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10-3

Time (sec)

Res

pons

e (m

)

Fig. 4. Impulse responses of the rotor system. —: DOF 5 (Y) first level control, - - -: DOF 6 (Z) first level control, y: DOF 5 (Y) robust

control, -.-: DOF 6 (Z) robust control.

H.-C. Yu et al. / Mechanical Systems and Signal Processing 21 (2007) 334–347 339

step responses of the controlled system, respectively, when the feedback gains in the first level control arereduced by half. Fig. 12 shows the impulse responses of the controlled system when the rotor speed has a largevariation from that of the original design. As can be seen, the controlled system is robust even with such largevariations of the rotor parameter.

Case C: In Figs. 13 and 14 are shown the responses at DOF 5 and DOF 6, respectively, when there is asudden disk mass loss of 1.347 103 kgm after 0.1 s in operation. In this case, the system is subjected to a

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1 1.2-20

-15

-10

-5

0

5

10

15

20

Time (sec)

Act

ive

inpu

t for

ce (

N)

Fig. 5. Control input forces for the rotor system subjected to a unit impulse. —: DOF 1(Y), - - -: DOF 2(Z), y: DOF 9(Y), -.-: DOF

10(Z).

0 0.5 1 1.5-6

-4

-2

0

2

4

6

8x 10-5

Time (sec)

Res

pons

e (m

)

Fig. 6. Dynamic responses of the rotor system subjected to step disturbance with duration of 0.2 s. —: DOF 5 (Y) first level control,

- - -: DOF 6 (Z) first level control, y: DOF 5 (Y) robust control, -.-: DOF 6 (Z) robust control.

H.-C. Yu et al. / Mechanical Systems and Signal Processing 21 (2007) 334–347340

sudden eccentric excitation. As can be seen, the robust control is capable of alleviating the system vibrationwhen unexpected excitation is present.

For the analyses described in Case A, the robust control applied in the second level provides a significantimprovement over the first level control. Note that the uncontrolled rotor system possesses repeated rigid bodymodes and hence is inherently unstable. Because the material damping is neglected and the higher modes are

ARTICLE IN PRESS

0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Act

ive

inpu

t for

ce (

N)

Fig. 7. Control input forces for the rotor system subjected to step disturbance with duration of 0.2 s. —: DOF 1(Y), - - -: DOF 2(Z),

y: DOF 9(Y), -.-: DOF 10 (Z).

0 0.5 1 1.5-8

-6

-4

-2

0

2

4

6

8

10x 10-4

Time (sec)

Res

pons

e (m

)

Fig. 8. Impulse responses of the rotor system at the mid-span with disk mass variation. —: DOF 5 (Y) original disk mass, - - -: DOF 6 (Z)

original disk mass, y: DOF 5 (Y) disk mass doubled, -.-: DOF 6 (Z) disk mass doubled.


not included in the controller design, the presence of higher mode residual vibration for impulse response isnot unexpected. The residual vibration is insignificant and cannot destabilise the system based on the robustcontrol characteristics. Note that the dominant modes contributing to the response are the first and the secondmodes, i.e. 5.532 and 7.141Hz, corresponding to the motions in the horizontal and vertical directions,respectively.

For the studies on the effects of parameter variation as presented in Case B, the step response is less sensitivethan the impulse response when the disk mass variation is present. When the disk mass is doubled, the

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0 0.5 1 1.5-4

-2

0

2

4

6

8x 10-5

Time (sec)

Res

pons

e (m

)

Fig. 9. Step responses of the rotor system at the mid-span with disk mass variation. —: DOF 5 (Y) original disk mass, - - -: DOF 6 (Z)

original disk mass, y: DOF 5 (Y) disk mass doubled, -.-: DOF 6 (Z) disk mass doubled.

0 0.5 1 1.5-1

-0.5

0

0.5

1

1.5x 10-3

Time (sec)

Res

pons

e (m

)

Fig. 10. Impulse responses of the rotor system at the mid-span with feedback gain variation in the first level control. —: DOF 5 (Y)

original gain, - - -: DOF 6 (Z) original gain, y: DOF 5 (Y) 50% of the original gain, -.-: DOF 6 (Z) 50% of the original gain.


response frequency is decreased, as may be expected. As can be seen in Figs. 10 and 11, the reduction of thefeedback gains in the first level control results in greater dynamic response with a lower frequency ofoscillation. This is to be expected since the situation is similar to the use of softer supports at the bearinglocations and the system becomes less rigid. The gyroscopic effect is more significant with a higher rotor speed.However, the present robust control design provides excellent vibration suppression performance even with a

ARTICLE IN PRESS

0 0.5 1 1.5-1

-0.5

0

0.5

1

1.5x 10-4

Time (sec)

Res

pons

e (m

)

Fig. 11. Step responses of the rotor system at the mid-span with feedback gain variation in the first level control. —: DOF 5 (Y) original

gain, - - -: DOF 6 (Z) original gain, y: DOF 5 (Y) 50% of the original gain, -.-: DOF 6 (Z) 50% of the original gain.

0 0.5 1 1.5-8

-6

-4

-2

0

2

4

6

8

10x 10-4

Time (sec)

Res

pons

e (m

)

Fig. 12. Impulse responses of the rotor system at the mid-span with speed variation. —: DOF 5 (Y) 4000 rpm, - - -: DOF 6 (Z) 4000 rpm,

y: DOF 5 (Y) 9500 rpm, -.-: DOF 6 (Z) 9500 rpm.


large amount of speed variation, as shown in Fig. 12. Note that the controlled response curves for the casesof DOF 5 (Y) 4000 rpm and DOF 5 (Y) 9500 rpm almost overlap with each other, so do the curves forthe cases of DOF 6 (Z) 4000 rpm and DOF 6 (Z) 9500 rpm. This indicates the present control system isinsensitive to such a speed variation with more than doubled speed change unknown to the control system atthe design stage.

ARTICLE IN PRESS

Fig. 13. Dynamic responses of the rotor system at the mid-span with sudden loss of disk mass. Upper: DOF 5 (Y) first level control,

Lower: DOF 5 (Y) robust control.

Fig. 14. Dynamic responses of the rotor system at the mid-span with sudden loss of disk mass. Upper: DOF 6 (Z) first level control,

Lower: DOF 6 (Z) robust control.


In Case C, the problem of a sudden loss of disk mass is examined. Eccentric excitation is present due to theabrupt mass loss, which is not known to the control system in the design stage. As demonstrated in Figs. 13and 14, the robust control system can react to such a sudden change and excessive vibration of the rotorsystem can be successfully suppressed. Note that the frequency of the residual steady-state vibration is66.667Hz, corresponding to the rotor speed of 4000 rpm.


The out of band modes may have detrimental effect on the system stability and performance of somefeedback control systems. Modes with significant contribution must be considered in the control systemdesign. In the present work, the control action synthesised from the control strategy is used to excite theoriginal full system rather than the reduced system. Therefore, the effect of out of band modes has been takeninto account and no appreciable contribution on the system response has been observed. The present modalcontrol approach can never destabilise the out of band modes because the modes are controlled independently.In most cases for structural vibration control, the lower modes are controlled. In this work, the first fourmodes are targeted for control. The out of band modes, i.e. the higher modes, have higher frequencies ofoscillations and hence are more difficult to excite. In addition, the damping tends to have a larger effect onhigher modes and the contribution of these modes are likely to be insignificant [23].

4. Conclusions

Robust modal control design of a magnetically suspended rotor has been presented in this work using thefinite element formulation. The original system is augmented using the direct output control in the first level,which removes the repeated rigid body modes of the uncontrolled system. In the second level design, a robustcontroller is implemented in the complex modal space. The vibration modes are controlled independently andhence for each mode controlled, the system matrix equation concerned is only of size 2 2, against 2n 2n inthe traditional coupled control approach. Significant reduction of the computational effort is evident byusing the present approach. It has been shown that the present control design is robust for systems withuncertainties and parameter variations, such as disk mass variation, feedback gains variation, speed variation,and sudden disk mass loss, etc.

Experimental work based on the present control strategy is currently undertaken, which requires theconstruction of the rotor, magnetic bearings, sensing apparatus, and the signal processing unit. A separateforthcoming article will be presented to detail such an experimental endeavor.

Acknowledgements

The authors are grateful for the support of the National Science Council (NSC), Taiwan, Republic ofChina, under the Grant no. NSC 91-2611-E-019-024.

Appendix A

Fig. A1 shows the rotor finite element, which consists of two nodes, with four degrees of freedom at eachnode. The nodal element displacement vector can be described as [24]

fqge ¼ ½V 1 W 1 B1 G1 V 2 W 2 B2 G2T . (A.1)

The displacement vector within the element is interpolated as

V

W

¼ ½Ntfqge, (A.2a)

B

G

¼ Nr½ q

e, (A.2b)

where

½Nt ¼N1 0 0 N2 N3 0 0 N4

0 N1 N2 0 0 N3 N4 0

" #, (A.3a)

½Nr ¼0 N1 N2 0 0 N3 N4 0

N1 0 0 N2 N3 0 0 N4

" #. (A.3b)

ARTICLE IN PRESS

Γ

Ω

d1

d2

d3

d4

d5

d6

d7

d8

Y

W

B

X

V

X α

Z

Fig. A1. The rotor finite element.


The shape functions shown in Eq. (A.3) are:

N1 ¼ 11

aða2 þ 12gÞð12gxþ 3ax2 2x3Þ,

N2 ¼1

aða2 þ 12gÞ½ða2 þ 6gÞax ð2a2 þ 6gÞx2 þ ax3,

N3 ¼1

aða2 þ 12gÞð12gxþ 3ax2 2x3Þ,

N4 ¼1

aða2 þ 12gÞ½6gaxþ ð6g a2Þx2 þ ax3 ðA:4Þ

and

N1 ¼1

aða2 þ 12gÞð6x2 6axÞ,

N2 ¼1

aða2 þ 12gÞ½a3 þ 12ga ð4a2 þ 12gÞxþ 3ax2,

N3 ¼1

aða2 þ 12gÞð6ax 6x2Þ,

N4 ¼1

aða2 þ 12gÞ½3ax2 ð2a2 12gÞx, ðA:5Þ

where

g EI

kGA, (A.6)

in which a is the element length, x the axial co-ordinate, E the Young’s modulus, I the area moment of inertia,G the shear modulus, A the cross-sectional area, k the shear coefficient. The above shape functions werederived earlier [25] for static analysis of beams including the effect of shearing deformations. The additional


effect of rotary inertia was considered as well in this work. The analysis including the effects of shearingdeformations and rotary inertia is known as the Timoshenko beam theory.

References

[1] C.R. Burrows, P.S. Keogh, R. Tasaltin, Closed-loop vibration control of flexible rotors—an experimental study, Proceedings of the

Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 207 (1) (1993) 1–17.

[2] P.S. Keogh, C. Mu, C.R. Burrows, Optimized design of vibration controllers for steady and transient excitation of flexible rotors,

Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 209 (3) (1995) 155–168.

[3] C.R. Knospe, R.W. Hope, S.J. Fedigan, R.D. Williams, Experiments in the control of unbalance response using magnetic bearings,

Mechatronics 5 (4) (1995) 385–400.

[4] Z. Yu, L.T. Meng, L.M. King, Electromagnetic bearing actuator for active vibration control of a flexible rotor, Proceedings of the

Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 212 (8) (1998) 705–716.

[5] P.S. Keogh, M.O.T. Cole, C.R. Burrows, Multi-state transient rotor vibration control using sampled harmonics, Journal of Vibration

and Acoustics, Transactions of the ASME 124 (2) (2002) 186–197.

[6] M.O.T. Cole, P.S. Keogh, C.R. Burrows, Control of multifrequency rotor vibration components, Proceedings of the Institution of

Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216 (2) (2002) 165–178.

[7] G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and appropriate inverses,

IEEE Transaction on Automatic Control AC-23 (1981) 301–320.

[8] J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State space solutions to standard H2 and HN control problems, IEEE

Transaction on Automatic Control 34 (8) (1989) 832–847.

[9] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, Macmillan Publishing Company, New York, 1992.

[10] K. Nonami, J.W. Wang, M. Sampei, T. Mita, Active vibration control of flexible rotors using HN control theory, rotating machinery

and vehicle dynamics, ASME DE-35 (1991) 85–91.

[11] M.O.T. Cole, P.S. Keogh, M.N. Sahinkaya, C.R. Burrows, Towards fault-tolerant active control of rotor-magnetic bearing systems,

Control Engineering Practice 12 (2004) 491–501.

[12] L. Meirovitch, Dynamics and Control of Structures, Wiley, New York, 1992.

[13] W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach, Springer, New York, 1998.

[14] L. Meirovitch, H. Baruh, H. Oz, A comparison of control techniques for large flexible systems, Journal of Guidance and Control 6 (4)

(1983) 302–310.

[15] Y.-H. Lin, C.-L. Chu, A new design for independent modal space control of general dynamic systems, Journal of Sound and

Vibration 180 (1995) 351–361.

[16] L. Meirovitch, H. Oz, Modal-space control of large flexible spacecraft possessing ignorable coordinates, Journal of Guidance and

Control 3 (6) (1980) 569–577.

[17] Y.-H. Lin, H.-C. Yu, Active modal control of a flexible rotor, Mechanical Systems and Signal Processing 18 (2004) 1117–1131.

[18] K. Zhou, J.C. Dole, Essentials of Robust Control, Prentice-Hall, Inc., New Jersey, 1998.

[19] W. Cui, K. Nonami, H. Nishimura, Experimental study on active vibration control of structures by means of HN control and H2

control, JSME Transactions Series C 37 (3) (1994) 462–467.

[20] K. Glover, J.C. Doyle, State space formulae for all stabilizing controllers that satisfying an HN-norm bound and relations to risk

sensitivity, Systems and Control Letters 11 (1988) 167–172.

[21] L. Meirovitch, H. Oz, Modal-space control of distributed gyroscopic systems, Journal of Guidance and Control 3 (2) (1980) 140–150.

[22] M.J. Balas, Active control of flexible systems, in: Proceedings of the AIAA Symposium on Dynamics and Control of Large Flexible

Spacecraft, Blacksburg, Va, 13–15 June 1977, pp. 217–236.

[23] L. Meirovitch, H. Baruh, Control of self-adjoint distributed-parameter systems, Journal of Guidance and Control 5 (1) (1982) 60–66.

[24] H.D. Nelson, The dynamics of rotor-bearing systems using finite elements, ASME Journal of Engineering for Industry 93 (2) (1976)

593–600.

[25] R. Narayanaswami, H.M. Adelman, Inclusion of transverse shear deformation in finite element displacement formulation, AIAA

Journal 12 (11) (1974) 1613–1614.

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Robust modal vibration suppression of a flexible rotor