Second order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. A regrettable consequence of this approach is the discarding of residual o-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper a method to utilise modal control using the decoupled second order matrix equations involving nonclassical damping is proposed. An example of modal control sucessfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components

Garvey, Seamus D.

Houlston, Paul Robert

Popov, Atanas A.

Second order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. A regrettable consequence of this approach is the discarding of residual off-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper a method to utilise modal control using the decoupled second order matrix equations involving non-classical damping is proposed. An example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components

Nottingham eTheses

Modal Control of Vibration in Rotating Machines and Other
Generally Damped Systems

Second order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. A regrettable consequence of this approach is the discarding of residual o-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper a method to utilise modal control using the decoupled second order matrix equations involving non-classical damping is proposed. An example of modal control sucessfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components

P. R. Houlston

S. D. Garvey

A. A. Popov

CiteSeerX

A.; Modal control of vibration in rotating machines and other generally damped systems

Nottingham ePrints

English

Second order matrix equations arise in the description of real dynamical systems. Tradi-tional modal control approaches utilise the eigenvectors of the undamped system to diago-nalise the system matrices. A regrettable consequence of this approach is the discarding of residual off-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper a method to utilise modal control using the decoupled second order matrix equations involving non-classical damping is proposed. An example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components

Houlston, Paul Robert and Garvey, Seamus D. and Popov, Atanas A. (2006) Modal Control of Vibration in Rotating Machines and Other Generally Damped Systems. In: ISMA 2006 International Conference on Noise and Vibration Engineering, 18-20 September 2006, Leuven, Belgium. Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/450/1/ISMA_V01.pdfCopyright and reuse: The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions.This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence.  For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdfA note on versions: The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.ukModal Control of Vibration in Rotating Machines and OtherGenerally Damped SystemsP.R. Houlston, S.D. Garvey and A.A. PopovApril 10, 2006AbstractSecond order matrix equations arise in the description of real dynamical systems. Traditionalmodal control approaches utilise the eigenvectors of the undamped system to diagonalise the systemmatrices. A regrettable consequence of this approach is the discarding of residual off-diagonal termsin the modal damping matrix. This has particular importance for systems containing skew-symmetryin the damping matrix which is entirely discarded in the modal damping matrix. In this paper amethod to utilise modal control using the decoupled second order matrix equations involving non-classical damping is proposed. An example of modal control sucessfully applied to a rotating systemis presented in which the system damping matrix contains skew-symmetric components.Keywords: modal control, second order systems, general damping, non-proportional damping, rotordy-namics1 IntroductionTraditional control approaches, such as pole placement methods [1], deal with the physical system infirst order state space form. The ambitions of this paper are to control the physical system in secondorder form. Very little literature is available in regards to direct second order control, see for example[2]. Many obvious advantages over first order control are available: 1.) Physical insight of the system ispreserved. 2.) Computational efficiency, since the dimension of the second order system is smaller thanthat of the state space form. 3.) Symmetry and structure of the systems can be preserved where desired.Many structural and dynamic systems are described by the second order equations of motionM0 q¨(t) +D0 q˙(t) +K0 q(t) = fphy(t) · (1)where M0,D0,K0 ∈ Rn×n are the system mass, damping and stiffness matrices respectively, q(t) ∈ Rnthe vector of physical coordinates and fphy(t) ∈ Rr the vector of applied forces. For the sake of brevitythis paper assumes that forces are available at all locations and as a consequence r = n.Modal control is a particular control method in which the physical response of a system is divided intomodes associated with their corresponding natural frequencies. A standard control approach is to move1the natural frequencies into a stable region. The essence of modal control is that since the eigenvectorsof a system do not contribute to the asympotic stability of a system then any effort expended on alteringthem represents wasted effort. This is the control approach utilised in this paper.Traditional modal control for second order systems such as the Independent Modal Space Control(IMSC) method [3] proposed by Meirovitch and Baruh utilise the mass normalised left and right eigen-vectors, ΦL and ΦR, to diagonalise the system matrices. The coordinate transformation q(t) = ΦRqm(t)is applied and the system matrices pre-multiplied by the transpose of the left eigenvectors, ΦLTFromΦLT M0 ΦR q¨m +ΦLT D0 ΦR q˙m +ΦLT K0 ΦR qm = ΦLT fphy · (2)one hasI q¨m + Γ q˙m +Λ2 qm = ΦLT fphy · (3)where qm(t) represents the modal coordinates of the system. For ease of reading the time script has beenremoved.The new damping matrix Γ is assumed diagonal with any remaining off-diagonal terms in the modaldamping matrix traditionally discarded [4]. However, for rotating systems involving substantial gyroscopicterms ignoring these terms is in effect ignoring the gyroscopic terms themselves. Thus, it is proposed hereto use the Structure Preserving Transformations (SPTs) developed by Garvey et al [5, 6] to diagonalisethe second order system matrices and decouple the system equations of motion without need to discardany terms involved in the description of the system.2 Structural Preserving TransformationsThe notion of the Lancaster Augmented Matrices (LAMs) are introduced here such that the system maybe represented in state space form. For a second order system there exists three LAMs which can beproduced by inspection to be,A0 = 0 K0K0 D0 , A1 = K0 00 −M0 , A2 = −D0 −M0−M0 0 · (4)The LAMs allow the second order system to be represented in a reduced formAk−1 qA −Ak q˙A = fA(3−k) k = 1, 2 · (5)The vectors qA and fA(3−k) may be definedqA := qq˙ fA1 := fphy0 fA2 := 0fphy · (6)2A Structural Preserving Transformation (SPT) is a coordinate transformation applied to the LAMsrepresenting a bijective mapping between linear systems. The specific nature of the transformation allowsthe preservation of the appropriate structure within the LAMs. The SPTs are defined simply by left andright 2n× 2n transformation matrices, TL and TR respectively, allowing the definitionTTL Ak TR = Bk ∀ k = 0, 1, 2 · (7)Thus the new LAMs are represented byBk containing the new second order systemmatricesK1,D1,M1.The structure of the SPTs can be shown to have the following formTL = FL −12 GL DT0 −GL MT0GL KT0 FL +12 GL DT0−1TR = FR −12 GR D0 −GR M0GR K0 FR +12 GR D0−1· (8)where FL,FR,GL,GR ∈ Rn×n are arbitrary pre-defined matrices subject to the necessary constraintFR GTL +GR FTL = 0 · (9)The SPTs can be shown to yield the relationship between the old and new coordinate sets throughthe use of filters. The modal system (qm, fmod) is related to the original system through the relationshipqm = U0 q+U1 q˙ (10)q˙m = U0 q˙+U1 q¨ (11)fmod = V0 fphy +V1 f˙phy (12)where[U0 U1]=[I 0]TR−1 (13)[VT0 VT1]=[I 0]TL (14)Evidently knowledge of the physical accelerations is required.3 Diagonalising Structural Preserving TransformationsWe wish to decouple the original equations of motion such that the new system matrices K1,D1 andM1are diagonal. It is possible to choose a non-unique SPT such that the entries in the new LAMs becomediagonal. Such an SPT is referred to as a diagonalising SPT (DSPT) and a 4 step process of calculatingthe DSPT is presented here.1. Calculate the left (ΨL) and right (ΨR) eigenvectors of reduced systemA0 qA −A1 q˙A ·32. Calculate the n single degree of freedom (SDOF) systems corresponding to conjugate eigenvaluepairs, λj(1,2) = α± iβ, found in part 1.dj = λj1 + λj2 , kj =(λj2 + λj1)2− (λj2 − λj1)24, mj = 1 · (15)j = 1, · · · , n.3. Knowing the new diagonal system matrices form the new LAMs B0 and B1 representing the newdiagonal system and calculate their corresponding left (ΘL) and right (ΘR) eigenvectors.4. Since the two reduced systems have identical Jordan form appropriate scaling of the eigenvectorsyields the following equalityΨTL A0 ΨR = Λ = ΘTL B0 ΘR ΨTL A1 ΨR = I = ΘTL B1 ΘR · (16)where Λ is the diagonal matrix of corresponding eigenvalues and I is the identity matrix. Thus wemay recognise that to get from the original LAM to the new LAM the following condition must besatisfied(ΘL−T ΨLT)A0(ΨR ΘR−1)= B0 · (17)thus TR =(ΨR ΘR−1)and TL =(ΨL ΘL−1).It may be noted that the above process for finding the diagonalising SPT only requires one eigenvaluesolution problem. The eigenvectors of the diagonal LAMs, ΘL and ΘR, have a sparse form such thattheir calculation is trivial.4 Independent Modal ControlTo facilitate true independent modal control the modal equations of motion must be decoupled bothexternally and internally [7]. We have so far shown how to decouple the unforced equations of motionbut the diagonalised system matrices remain coupled by the control forces unless the controller is designedindependently such that the controller matrix remains decoupled. In practice this means that the forcecontroller must be designed in the modal space. We can thus define the modal equations of motion asM1 q¨m +D1 q˙m +K1 qm = fmod · (18)with K1,D1,M1 ∈ Rn×n the diagonal modal system matrices and qm ∈ Rn the modal coordinates.Equation (18) represents n single degree of freedom (SDOF) systems corresponding to each mode ofvibration. It is possible to use proportional-derivative control to directly affect the modal stiffness anddamping properties of these modes. A controller of this form is introduced4fmod = Gk qm +Gd q˙m · (19)Gk and Gd represent the diagonal modal stiffness and damping gains matrices. Direct additon to themodal damping and stiffness matrices represents direct pole placement and has the advantage of beingable to directly affect the poles of the system.In general as many modes can be controlled as actuators available. As previously stated for thepurpose of this paper the number of actuators is set to the number of modelled modes without loss ofgenerality. For conventional second order control the modal force can be typically converted back into thephysical domain fairly easily as illustrated by Baz and Poh [8]. For the SPT approach we have alreadydefined the left filter and can see that the physical and modal forces are related by the relationshipfmod = V0 fphy +V1 f˙phy · (20)We can rearrange equation (20) to give the physical force in regards to the modal forcef˙phy = V−11 (fmod −V0 fphy) · (21)Since the modal filter illustrated by equation (21) represents a first order filter a necessary requirementis for the real eigenvalue components of V−11 V0 > 0 for the filter to be stable.5 Numerical ExampleAs a numerical example a finite element model of a rotor-disc system is considered with four degrees offreedom at each node (2 translational, 2 torsional). The rotor-disc system is illustrated in figure 1.Figure 1: Example 1, Rotor-Disc systemThe system is constructed from steel with Young’s modulus, E = 200 GPa and density ρ = 7800kg/m3. The model is split into 13 equal-length elements of 0.1m and the discs have dimensions5Disc Disc 1 Disc 2 Disc 3Node 3 6 11Thickness (m) 0.05 0.05 0.06Inner diameter (m) 0.10 0.10 0.10Outer diameter (m) 0.24 0.40 0.40The bearings at each end of the rotor system are deliberately non-symmetric in the x-y directionswith stiffness and damping propertiesBearing Bearing 1 Bearing 2Stiffness Kxx (MN/m) 50 50Stiffness Kyy (MN/m) 70 70Stiffness Dxx (N/m/s) 500 500Stiffness Dyy (N/m/s) 700 700Control forces can be applied at node 8 in the x and y-directions and similarly the displacements inthe x-direction at this node are observed. For computational ease guyan reduction [9] is used to reducethe model to 6 degrees of freedom. The system is operated at 2,500 rpm.As many modes can be controlled as actuators are available, thus the model allows for 2 modes tobe controlled. It is decided to control the first two modes of vibration since these dominate the systemresponse.0 1 2 3 4 5 6 7 8 9 10−0.02−0.015−0.01−0.00500.0050.010.0150.020.025time, sphy displacementFigure 2: Example 1 SPT response to initial conditions: control offThe single degree of freedom systems corresponding to the first two modes in modal space are6q¨m1 + 0.37850 q˙m1 + 1.4467× 105 qm1 = fm1 (22)q¨m2 + 0.32708 q˙m2 + 1.5772× 105 qm2 = fm2 (23)Optimal control is used to minimise the modal kinetic and potential energies such that controllergains areGk = 4.9999 0 0 · · · 00 4.9999 0 · · · 0 , Gd = 4.1096 0 0 · · · 00 4.1570 0 · · · 0 (24)The response of the system with the controller on is illustrated in figure 3.0 1 2 3 4 5 6 7 8 9 10−0.02−0.015−0.01−0.00500.0050.010.0150.02time, sphy displacementFigure 3: Example 1 SPT response to initial conditions: control onAs expected the response of the system decays much faster than that for the uncontrolled systemwith the displacement converging to zero much more rapidly. This is due to tageting the first two modesof vibration of the system which dominate the system response. The modal control technique is indeedsucessfully applied to bring the system under control.6 ConclusionsIt has been shown in this paper how to apply modal control to non-classically damped systems withoutthrowing away system information. It has been demonstrated through examples that individual modescan be controlled.7The premise of this paper is to introduce possible new methods into the area of rotating machinerywhere skew-symmetry and gyroscopic coupling can be found in the system damping matrices. Conven-tional techniques maintain that skew-symmetry be ignored for the techniques to be usable.Usually, systems require reduction in size due to numerical considerations. Traditional Guyan re-duction models do not take into account damping properties. Alternative methods such as balancedtruncation [4] traditionally place the system into state space form before reduction, thus destroying thesecond order properties of the system. Few methods have been developed to reduce the models in sizefor second order systems. Currently the methods for reducing second order systems appear to involvethe use of state space form to balance the grammians [10]. It would thus be beneficial to develop secondorder model reduction methods that take into account damping whilst preserving second order form.7 AcknowledgementsP. Houlston would like to acknowledge the support of the Engineering and Physical Sciences ResearchCouncil (EPSRC) and Rolls-Royce PLC.References[1] Kautsky J., Nichols N.K. and Van Dooren V., Robust Pole Assignment in Linear State Feedback,International Journal of Control, 1985, 41(5), pp. 1129-1155.[2] A.M. Diwekar and R.K. Yedavalli (1999): Robust Controller Design for Matrix Second Order Systemswith Structured Uncertainty. IEEE Transactions Automatic Control. 44(2), pp. 401-405[3] Meirovitch L. and Baruh H., Control of Self-Adjoint Distributed Parameter Systems, Journal ofGuidance, Control and Dynamics, 1982, 5(1), pp. 60-66.[4] W.K. Gawronski (1998): Dynamics and Control of Structures: A Modal Approach. Springer-VerlagNew York Inc.[5] S.D. Garvey, U. Prells and M.I. Friswell (2001): Diagonalising coordinate transformations for systemswith general viscous damping. 19th International Modal Analysis Conference, Orlando, 1(1), pp.622-627[6] S.D. Garvey, M.I. Friswell and U. Prells (2002): Coordinate Transformations for General SecondOrder Systems, Pt. I: General Transformations. Journal of Sound and Vibration, 258(5), pp. 885-909.[7] Meirovitch L., Dynamics and Control of Structures, Wiley, New York, 1990.[8] Baz. A, S. Poh and Studer P., Modified Independent Modal Space Control Method for Active Controlof Flexible Systems, Proceedings of the Institute of Mechanical Engineers Part C, 1989, 203(2), pp.103-112.8[9] Guyan R.J., Reduction of Stiffness and Mass Matrices, AIAA Journal, 1964, 3(2), pp. 380.[10] Benner P., Mehrmann V. and Sorensen D., Dimension Reduction of Large Scale Systems, Springer-Verlag Berlin, 2005.9

Modal Control of Vibration in Rotating Machines and Other Generally Damped Systems

Name not available

Modal Control of Vibration in Rotating Machines and OtherGenerally Damped SystemsP.R. Houlston, S.D. Garvey and A.A. PopovApril 10, 2006AbstractSecond order matrix equations arise in the description of real dynamical systems. Traditionalmodal control approaches utilise the eigenvectors of the undamped system to diagonalise the systemmatrices. A regrettable consequence of this approach is the discarding of residual off-diagonal termsin the modal damping matrix. This has particular importance for systems containing skew-symmetryin the damping matrix which is entirely discarded in the modal damping matrix. In this paper amethod to utilise modal control using the decoupled second order matrix equations involving non-classical damping is proposed. An example of modal control sucessfully applied to a rotating systemis presented in which the system damping matrix contains skew-symmetric components.Keywords: modal control, second order systems, general damping, non-proportional damping, rotordy-namics1 IntroductionTraditional control approaches, such as pole placement methods [1], deal with the physical system infirst order state space form. The ambitions of this paper are to control the physical system in secondorder form. Very little literature is available in regards to direct second order control, see for example[2]. Many obvious advantages over first order control are available: 1.) Physical insight of the system ispreserved. 2.) Computational efficiency, since the dimension of the second order system is smaller thanthat of the state space form. 3.) Symmetry and structure of the systems can be preserved where desired.Many structural and dynamic systems are described by the second order equations of motionM0 q¨(t) + D0 q˙(t) + K0 q(t) = fphy(t) · (1)where M0,D0,K0 ∈ Rn×n are the system mass, damping and stiffness matrices respectively, q(t) ∈ Rnthe vector of physical coordinates and fphy(t) ∈ Rr the vector of applied forces. For the sake of brevitythis paper assumes that forces are available at all locations and as a consequence r = n.Modal control is a particular control method in which the physical response of a system is divided intomodes associated with their corresponding natural frequencies. A standard control approach is to move1the natural frequencies into a stable region. The essence of modal control is that since the eigenvectorsof a system do not contribute to the asympotic stability of a system then any effort expended on alteringthem represents wasted effort. This is the control approach utilised in this paper.Traditional modal control for second order systems such as the Independent Modal Space Control(IMSC) method [3] proposed by Meirovitch and Baruh utilise the mass normalised left and right eigen-vectors, ΦL and ΦR, to diagonalise the system matrices. The coordinate transformation q(t) = ΦR qm(t)is applied and the system matrices pre-multiplied by the transpose of the left eigenvectors, ΦLTFromΦLT M0 ΦR q¨m + ΦLT D0 ΦR q˙m + ΦLT K0 ΦR qm = ΦLT fphy · (2)one hasI q¨m + Γ q˙m + Λ2 qm = ΦLT fphy · (3)where qm(t) represents the modal coordinates of the system. For ease of reading the time script has beenremoved.The new damping matrix Γ is assumed diagonal with any remaining off-diagonal terms in the modaldamping matrix traditionally discarded [4]. However, for rotating systems involving substantial gyroscopicterms ignoring these terms is in effect ignoring the gyroscopic terms themselves. Thus, it is proposed hereto use the Structure Preserving Transformations (SPTs) developed by Garvey et al [5, 6] to diagonalisethe second order system matrices and decouple the system equations of motion without need to discardany terms involved in the description of the system.2 Structural Preserving TransformationsThe notion of the Lancaster Augmented Matrices (LAMs) are introduced here such that the system maybe represented in state space form. For a second order system there exists three LAMs which can beproduced by inspection to be,A0 = 0 K0K0 D0 , A1 = K0 00 −M0 , A2 = −D0 −M0−M0 0 · (4)The LAMs allow the second order system to be represented in a reduced formAk−1 qA −Ak q˙A = fA(3−k) k = 1, 2 · (5)The vectors qA and fA(3−k) may be definedqA := qq˙ fA1 := fphy0 fA2 := 0fphy · (6)2A Structural Preserving Transformation (SPT) is a coordinate transformation applied to the LAMsrepresenting a bijective mapping between linear systems. The specific nature of the transformation allowsthe preservation of the appropriate structure within the LAMs. The SPTs are defined simply by left andright 2n× 2n transformation matrices, TL and TR respectively, allowing the definitionTTL Ak TR = Bk ∀ k = 0, 1, 2 · (7)Thus the new LAMs are represented by Bk containing the new second order system matrices K1,D1,M1.The structure of the SPTs can be shown to have the following formTL = FL −12 GL DT0 −GL MT0GL KT0 FL +12 GL DT0−1TR = FR −12 GR D0 −GR M0GR K0 FR +12 GR D0−1· (8)where FL,FR,GL,GR ∈ Rn×n are arbitrary pre-defined matrices subject to the necessary constraintFR GTL + GR FTL = 0 · (9)The SPTs can be shown to yield the relationship between the old and new coordinate sets throughthe use of filters. The modal system (qm, fmod) is related to the original system through the relationshipqm = U0 q + U1 q˙ (10)q˙m = U0 q˙ + U1 q¨ (11)fmod = V0 fphy + V1 f˙phy (12)where[U0 U1]=[I 0]TR−1 (13)[VT0 VT1]=[I 0]TL (14)Evidently knowledge of the physical accelerations is required.3 Diagonalising Structural Preserving TransformationsWe wish to decouple the original equations of motion such that the new system matrices K1,D1 and M1are diagonal. It is possible to choose a non-unique SPT such that the entries in the new LAMs becomediagonal. Such an SPT is referred to as a diagonalising SPT (DSPT) and a 4 step process of calculatingthe DSPT is presented here.1. Calculate the left (ΨL) and right (ΨR) eigenvectors of reduced systemA0 qA −A1 q˙A ·32. Calculate the n single degree of freedom (SDOF) systems corresponding to conjugate eigenvaluepairs, λj(1,2) = α± iβ, found in part 1.dj = λj1 + λj2 , kj =(λj2 + λj1)2− (λj2 − λj1)24, mj = 1 · (15)j = 1, · · · , n.3. Knowing the new diagonal system matrices form the new LAMs B0 and B1 representing the newdiagonal system and calculate their corresponding left (ΘL) and right (ΘR) eigenvectors.4. Since the two reduced systems have identical Jordan form appropriate scaling of the eigenvectorsyields the following equalityΨTL A0 ΨR = Λ = ΘTL B0 ΘR ΨTL A1 ΨR = I = ΘTL B1 ΘR · (16)where Λ is the diagonal matrix of corresponding eigenvalues and I is the identity matrix. Thus wemay recognise that to get from the original LAM to the new LAM the following condition must besatisfied(ΘL−T ΨLT)A0(ΨR ΘR−1)= B0 · (17)thus TR =(ΨR ΘR−1)and TL =(ΨL ΘL−1).It may be noted that the above process for finding the diagonalising SPT only requires one eigenvaluesolution problem. The eigenvectors of the diagonal LAMs, ΘL and ΘR, have a sparse form such thattheir calculation is trivial.4 Independent Modal ControlTo facilitate true independent modal control the modal equations of motion must be decoupled bothexternally and internally [7]. We have so far shown how to decouple the unforced equations of motionbut the diagonalised system matrices remain coupled by the control forces unless the controller is designedindependently such that the controller matrix remains decoupled. In practice this means that the forcecontroller must be designed in the modal space. We can thus define the modal equations of motion asM1 q¨m + D1 q˙m + K1 qm = fmod · (18)with K1,D1,M1 ∈ Rn×n the diagonal modal system matrices and qm ∈ Rn the modal coordinates.Equation (18) represents n single degree of freedom (SDOF) systems corresponding to each mode ofvibration. It is possible to use proportional-derivative control to directly affect the modal stiffness anddamping properties of these modes. A controller of this form is introduced4fmod = Gk qm + Gd q˙m · (19)Gk and Gd represent the diagonal modal stiffness and damping gains matrices. Direct additon to themodal damping and stiffness matrices represents direct pole placement and has the advantage of beingable to directly affect the poles of the system.In general as many modes can be controlled as actuators available. As previously stated for thepurpose of this paper the number of actuators is set to the number of modelled modes without loss ofgenerality. For conventional second order control the modal force can be typically converted back into thephysical domain fairly easily as illustrated by Baz and Poh [8]. For the SPT approach we have alreadydefined the left filter and can see that the physical and modal forces are related by the relationshipfmod = V0 fphy + V1 f˙phy · (20)We can rearrange equation (20) to give the physical force in regards to the modal forcef˙phy = V−11 (fmod −V0 fphy) · (21)Since the modal filter illustrated by equation (21) represents a first order filter a necessary requirementis for the real eigenvalue components of V−11 V0 > 0 for the filter to be stable.5 Numerical ExampleAs a numerical example a finite element model of a rotor-disc system is considered with four degrees offreedom at each node (2 translational, 2 torsional). The rotor-disc system is illustrated in figure 1.Figure 1: Example 1, Rotor-Disc systemThe system is constructed from steel with Young’s modulus, E = 200 GPa and density ρ = 7800kg/m3. The model is split into 13 equal-length elements of 0.1m and the discs have dimensions5Disc Disc 1 Disc 2 Disc 3Node 3 6 11Thickness (m) 0.05 0.05 0.06Inner diameter (m) 0.10 0.10 0.10Outer diameter (m) 0.24 0.40 0.40The bearings at each end of the rotor system are deliberately non-symmetric in the x-y directionswith stiffness and damping propertiesBearing Bearing 1 Bearing 2Stiffness Kxx (MN/m) 50 50Stiffness Kyy (MN/m) 70 70Stiffness Dxx (N/m/s) 500 500Stiffness Dyy (N/m/s) 700 700Control forces can be applied at node 8 in the x and y-directions and similarly the displacements inthe x-direction at this node are observed. For computational ease guyan reduction [9] is used to reducethe model to 6 degrees of freedom. The system is operated at 2,500 rpm.As many modes can be controlled as actuators are available, thus the model allows for 2 modes tobe controlled. It is decided to control the first two modes of vibration since these dominate the systemresponse.0 1 2 3 4 5 6 7 8 9 10−0.02−0.015−0.01−0.00500.0050.010.0150.020.025time, sphy displacementFigure 2: Example 1 SPT response to initial conditions: control offThe single degree of freedom systems corresponding to the first two modes in modal space are6q¨m1 + 0.37850 q˙m1 + 1.4467× 105 qm1 = fm1 (22)q¨m2 + 0.32708 q˙m2 + 1.5772× 105 qm2 = fm2 (23)Optimal control is used to minimise the modal kinetic and potential energies such that controllergains areGk = 4.9999 0 0 · · · 00 4.9999 0 · · · 0 , Gd = 4.1096 0 0 · · · 00 4.1570 0 · · · 0 (24)The response of the system with the controller on is illustrated in figure 3.0 1 2 3 4 5 6 7 8 9 10−0.02−0.015−0.01−0.00500.0050.010.0150.02time, sphy displacementFigure 3: Example 1 SPT response to initial conditions: control onAs expected the response of the system decays much faster than that for the uncontrolled systemwith the displacement converging to zero much more rapidly. This is due to tageting the first two modesof vibration of the system which dominate the system response. The modal control technique is indeedsucessfully applied to bring the system under control.6 ConclusionsIt has been shown in this paper how to apply modal control to non-classically damped systems withoutthrowing away system information. It has been demonstrated through examples that individual modescan be controlled.7The premise of this paper is to introduce possible new methods into the area of rotating machinerywhere skew-symmetry and gyroscopic coupling can be found in the system damping matrices. Conven-tional techniques maintain that skew-symmetry be ignored for the techniques to be usable.Usually, systems require reduction in size due to numerical considerations. Traditional Guyan re-duction models do not take into account damping properties. Alternative methods such as balancedtruncation [4] traditionally place the system into state space form before reduction, thus destroying thesecond order properties of the system. Few methods have been developed to reduce the models in sizefor second order systems. Currently the methods for reducing second order systems appear to involvethe use of state space form to balance the grammians [10]. It would thus be beneficial to develop secondorder model reduction methods that take into account damping whilst preserving second order form.7 AcknowledgementsP. Houlston would like to acknowledge the support of the Engineering and Physical Sciences ResearchCouncil (EPSRC) and Rolls-Royce PLC.References[1] Kautsky J., Nichols N.K. and Van Dooren V., Robust Pole Assignment in Linear State Feedback,International Journal of Control, 1985, 41(5), pp. 1129-1155.[2] A.M. Diwekar and R.K. Yedavalli (1999): Robust Controller Design for Matrix Second Order Systemswith Structured Uncertainty. IEEE Transactions Automatic Control. 44(2), pp. 401-405[3] Meirovitch L. and Baruh H., Control of Self-Adjoint Distributed Parameter Systems, Journal ofGuidance, Control and Dynamics, 1982, 5(1), pp. 60-66.[4] W.K. Gawronski (1998): Dynamics and Control of Structures: A Modal Approach. Springer-VerlagNew York Inc.[5] S.D. Garvey, U. Prells and M.I. Friswell (2001): Diagonalising coordinate transformations for systemswith general viscous damping. 19th International Modal Analysis Conference, Orlando, 1(1), pp.622-627[6] S.D. Garvey, M.I. Friswell and U. Prells (2002): Coordinate Transformations for General SecondOrder Systems, Pt. I: General Transformations. Journal of Sound and Vibration, 258(5), pp. 885-909.[7] Meirovitch L., Dynamics and Control of Structures, Wiley, New York, 1990.[8] Baz. A, S. Poh and Studer P., Modified Independent Modal Space Control Method for Active Controlof Flexible Systems, Proceedings of the Institute of Mechanical Engineers Part C, 1989, 203(2), pp.103-112.8[9] Guyan R.J., Reduction of Stiffness and Mass Matrices, AIAA Journal, 1964, 3(2), pp. 380.[10] Benner P., Mehrmann V. and Sorensen D., Dimension Reduction of Large Scale Systems, Springer-Verlag Berlin, 2005.9

Control of Self-Adjoint Distributed Parameter Systems,

Coordinate Transformations for General Second Order Systems, Pt. I: General Transformations.

Diagonalising coordinate transformations for systems with general viscous damping.

Dimension Reduction of Large Scale Systems, SpringerVerlag

Dynamics and Control of Structures,

Dynamics and Control of Structures: A Modal Approach.

Modified Independent Modal Space Control Method for Active Control of Flexible Systems,

Reduction of Stiffness and Mass Matrices,

Robust Controller Design for Matrix Second Order Systems with Structured Uncertainty.

Robust Pole Assignment in Linear State Feedback,

http://eprints.nottingham.ac.uk/453/1/SPT_IMSC_Reviewed_V2.pdf

Modal Control of Vibration in Rotating Machines and Other Generally Damped Systems

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