Model Order Reduction of Rotordynamical Systems · reduced using a suitable reduction method. For instance, for large-sized systems with Krylov-subspace methods. The resulting subsystems

Model Order Reduction of RotordynamicalSystems

Kristin Krenek ∗, Rudy Eid ∗∗, Boris Lohmann ∗∗

∗ Siemens AG, Industry Sec., DT LD DW EN3, Nonnendammallee 72,13629 Berlin, Germany (e-mail: [email protected]).

∗∗ Lehrstuhl fur Regelungstechnik, Technische Universitat Munchen,Boltzmannstr. 15, 85748 Garching, Germany

(e-mail: eid,[email protected])

Abstract: This paper presents an approach for model order reduction of linear rotordynamicalsystems resulting from the accurate modeling of large machines with a rotor. Automaticallypartitioning the overall model into decoupled sub-models, describing the main components,namely, rotor, bearing and housing, allows these to be separately considered, and if necessary,reduced using a suitable reduction method. For instance, for large-sized systems with Krylov-subspace methods. The resulting subsystems are then re-coupled to deliver a reduced-ordermodel of the overall machine which can be simulated far faster then when compared to theoriginal one. As an example, the method presented here is applied to reduce a simplified finite-element model of a gas compressor drive.

Keywords: Model Order Reduction; Rotordynamics; Electrical machines; Reduced-ordermodels; Decoupling.

1. INTRODUCTION

Today, large rotordynamical machines are widely used innumerous industrial sectors. Ship’s engines, large millswith a diameter of several meters, high-power electricalgenerators, and wind turbines are just a few examples.At the Siemens ”Dynamowerk Berlin” (DW), where suchcustomized large drives are engineered and manufactured,one of the main tasks is to analyze the designed machineswhose dynamics are mainly influenced by that of therotor. Until just recently, this suggested that only therotor be modeled and simulated under different operatingconditions. Lately, in order to be able to produce morepowerful, reliable and lighter drives, the simulation of thecomplete machine, or even of the machine together withthe surroundings, e. g. the base on which it is mounted,is required. Accurate modeling of these systems, mostlyrealized using the Finite Element Method (FEM) leads toparametric large-scale dynamical systems with more than250,000 state variables, involving the rotational speed ofthe rotor as a parameter in the damping matrix. Hence,the computational cost of solving the underlying systemof equations is very expensive, especially if several modelshave to be coupled and solved simultaneously. For specificapplications, this can take up to three weeks.

Model Order Reduction (MOR) techniques offer a solutionto this problem, by making it possible to generate accuratereduced-order models that are able to approximate thebehavior of the original large-scale ones, while being muchfaster in the required simulation time. Thus, by gener-ating appropriate reduced-order models of the differentcomponents of a system or at least for some of them,the simulation of the complete system can be performedvery efficiently. For the class of large-scale systems (a few

hundred thousand), Krylov-subspace methods have provenlately to be one of the leading projection-based approaches(Antoulas (2005); Freund (2003); Grimme (1997)). Thesenumerically efficient methods are based on matching someof the first coefficients of the Taylor series expansion of thetransfer functions (the so-called moments) of the originaland reduced models. Recently, several studies tried to gen-eralize this approach to parametric systems (Daniel et al.(2004); Panzer et al. (2010); Peng et al. (2005)). However,due to several disadvantages, it is still too early to considerthat the problem of model reduction of parametric systemshas been solved.

By taking a closer look at the generated model of the rotat-ing machines being considered (Zienkiewicz et al. (2005);Huebner et al. (2001)), it can be observed that: i) the sub-model of the housing and of the baseplate, on which themachine is mounted, is by far the largest, and ii) only thesub-model describing the rotor is a parametric dynamicalsystem with a relatively low order (Yamamoto and Ishida(2001)). Accordingly, the new approach presented in thispaper consists of partitioning the automatically generatedFE-model into three sub-models, namely, rotor, bearingand housing, and then appropriately reducing that of thehousing so that the resulting subsystem can be re-coupledto the other two. This results in a reduced-order model ofthe overall machine which can be simulated within a muchshorter time compared to the original one, but which andstill allows the rotational speed to be varied.

The paper is structured as follows: the preliminaries, themodeling of the main components of the rotating machinesat Siemens DW and a short overview on the Krylov-based model order reduction are presented in the nexttwo sections. They show the problem with reducing the

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

3581

Fig. 1. Lateral cross-section of an FE-model of a gas com-pressor drive showing the main parts of the machine:the rotor (blue), the model of the fluid bearing (red),the bearing housing (green) and the housing (gray).

parameter depending rotordynamical structures. The twomain results, the partitioning of the machine model intoseveral sub-models together with its automation and thereduction and re-coupling to a reduced overall model, arepresented in Sec. 4 and 5. Finally, the new approach isapplied to reduce the model of a gas compressor drive.

2. MODELING OF ROTORDYNAMICAL MACHINES

The large machines produced at Siemens DW are cus-tomized drives with a power up to 100 MW and a weightup to 500 t. Therefore, before production, every designeddrive must be subject to a thorough simulation underdifferent operating conditions e. g. rotational speeds of therotor. This is achieved by a complex FE-model involvingthe main parts of the machine, as illustrated in Figure 1.In this section, a short overview of the modeling of themachine’s main parts, namely, the housing, the bearingsand the rotor is presented.

2.1 Model of the housing

The housing of the machine consists of a large metal struc-ture that can be easily modeled using the finite elementmethod leading to a second-order system of differentialequations of the form

Mhzh(t) + Dhzh(t) + Khzh(t) = Jhuh(t)

yh(t) = Lhzh(t),(1)

where Mh, Dh, Kh ∈ Rn×n are constant matrices repre-senting the mass, damping, and stiffness matrices of thesystem, respectively. The vector of internal generalizedcoordinates is zh(t) ∈ Rn, uh(t) ∈ Rm corresponds tothe vector of external forces, Jh ∈ Rn×m is the inputmatrix, yh(t) ∈ Rp is the output measurement vector, andLh ∈ Rp×n is the output matrix. The resulting model istypically non-parametric and of large-scale with at least100,000 states.

2.2 Model of the rotor

The main idea behind modeling the rotor is to divide it intorotating cylindrical beam sections while considering the

Fig. 2. Illustration of the gyroscopic moment of a rotatingdisk mounted on a mass-less shaft.

gyroscopic moment generated when rotating (Yamamotoand Ishida (2001), Friswell et al. (2010)). This guaranteesa realistic description and therefore more accurate simula-tion results.

The gyroscopic moment describes the coupling betweenthe lateral rotation and the angle of the rotor elementcaused by an axial mass moment of inertia. As an example,the gyroscopic moment of a rotating disk mounted on amass-less shaft is shown in Figure 2. A rotation around thex-axis with ϕx = −Ω and a rotation ϕy along the y-axis,create a moment Mz around the z-axis leading to shaftbending. Hence, the gyroscopic moment Mz around thez-axis of the disk with the polar mass moment of inertiaΘx around the x-axis is

Mz = ΘxΩϕy. (2)

Please note that for the machines considered in thispaper, incorporating the gyroscopic moment in the modelis crucial for its accuracy, as for even small tangentialdeviations of the large-sized rotor, large mass momentsof inertia are present.

By combining the gyroscopic moments for each rotor ele-ment in the gyroscopic matrix G(Ω), the rotor dynamicscan be described by a parametric second-order system ofthe form

Mtzt(t) + (Dt + Gt(Ω)) zt(t) + Ktzt(t) = Jtut(t)

yt(t) = Ltzt(t),(3)

where the structure of the system matrices are comparablewith Eq. (1). The resulting model has typically only therotational speed Ω as a parameter and does not consist ofmore than 300 second-order differential equations.

2.3 Model of the fluid bearings

The rotor is coupled to the housing through the bearings,whose type, arrangement, and number strongly dependson the size of the machine. For the drives produced at theSiemens DW, fluid bearings are the most commonly usedones, however with an increasing trend towards employingmagnetic bearings. As fluid bearings significantly influencethe dynamic interaction between rotor and housing, theywill be considered in this paper.

The fluid bearings are modeled in a separate specializedtool which includes the non-linearities and the characteris-tics of the oils being used (Glienicke et al. (1980), Athavaleand Hendricks (1996)) and provides suitable damping andstiffness matrices, so that the model of the fluid bearing


3582

can be replaced by that of a spring-damper device. One ofthe main characteristics of the resulting matrices is thatthey have coupling terms between the different degrees offreedom involved 1 leading generally to non-symmetricalsystem matrices Do and Ko.

Such a modeling approach of the bearing results in asystem where the rotor and the housing are coupled toeach other via the stiffness and the damping matrices ofthe substitute spring-damper model. This is reflected inthe layout of the overall system matrices K and D asshown below 2 for a simplified model

K =

kt11 kt12 0 0

kt21 kt22 + ko −ko 0

0 −ko kh11 + ko kh12

0 0 kh21kh22

rotor Ktbearinghousing Kh

(4)This exemplary stiffness matrix corresponds to a systemwith only two degrees of freedom for the rotor and forthe housing, which are coupled to each other through onedegree of freedom.

2.4 Overall model of the machine

Practically, it is not possible to generate each of the threesub-models separately. The FEM software tool delivers thefollowing model of the complete machine

Mz(t) + (D + G(Ω)) z(t) + Kz(t) = Ju(t)

y(t) = Lz(t).(5)

which consists of few hundred thousands of second-orderdifferential equations. It incorporates all of the three sub-models and their dynamic interactions, however, in a cou-pled manner that does not allow a straightforward decou-pling. Hence, there is a need for an automatic algorithmthat first partitions the system (5) into the sub-modelslisted above, allowing, if necessary, to reduce them sepa-rately, and then re-couple the resulting models to conducta complete system simulation (see Sec. 4 and 5).

3. KRYLOV-BASED MODEL ORDER REDUCTION

Consider the linear time-invariant multi-input and multi-output system

Ex(t) = Ax(t) + Bu(t),

y(t) = Cx(t),(6)

of order n ∈ N, where E, A ∈ Rn×n, B ∈ Rn×m,C ∈ Rp×n are matrices with constant coefficients, u(t) ∈Rm, y(t) ∈ Rp, and x(t) ∈ Rn, respectively, the input,output and state vectors of the system. The transfermatrix of Eq. (6),

H(s) = C(sE−A)−1B. (7)

admits the infinite Taylor series’ expansion about s01 It is assumed here that the bearing does not move in the directionof the rotor’s axis.2 The layout of the matrix D is similar.

H(s) =

∞∑i=0

−C((A− s0E)−1E)i(A− s0E)−1B(s− s0)i

=

∞∑i=0

mis0(s− s0)i

The aim of Krylov-based model reduction is to find areduced model of order q n, whose first few momentsmis0 around a certain expansion point s0 match those of

the original one (Freund (2003)). This approach is alsoknown as moment matching.

A numerically efficient possibility to calculate such areduced-order model is applying a projection to the origi-nal model,

Er︷︸︸︷WTEV xr(t) =

Ar︷︸︸︷WTAVxr(t) +

Br︷︸︸︷WTBu(t),

y(t) = CV︸︷︷︸Cr

xr(t),(8)

by means of the so-called projection matrices V and W.For the choice of these matrices, the block Krylov subspaceKS , defined in e.g. (Grimme (1997)) is employed,

KS(A,B) = spanB,AB, · · · ,AQ−1B,where A ∈ Rn×n, B ∈ Rn×m, and S = Q ·m. Specifically,if the projection matrices are chosen such that

Kq1((A− s0E)−1E, (A− s0E)−1B

)⊆ colspan(V),

Kq2((A− s0E)−TET , (A− s0E)−TCT

)⊆ colspan(W),

then the first(q1m + q2

p

)moments of the original and

reduced models match, and the procedure is called a two-sided Krylov method. In the so-called one-sided method,only one Krylov subspace is used with the common choiceW = V and only q1

m or q2p moments match (Grimme

(1997)). Note that for the matrices V and W to haveappropriate dimensions, the order of the reduced systemshould be a common multiple of the number of inputs andoutputs.

For the numerical computation of the matrices V and Wthe Lanczos, Arnoldi and two-sided Arnoldi algorithmsand their numerous improvements and modified versionsare used. For more details see e. g. (Antoulas (2005)) andthe references therein.

4. PARTITIONING THE OVERALL MACHINEMODEL

As already discussed in Sec. 2, the overall machine modelconsists of the model of the rotor coupled to that of thehousing by means of the equivalent spring-damper modelof the fluid bearings. Hence, the aim now is to separatethe rotor from the housing while preserving the completedynamical characteristics of the overall model.

As it is free from coupling terms, the overall mass matrixM can be easily divided into a rotor and a housing part,Mt and Mh, respectively. It is also straightforward toassign the non-zero elements of the overall gyroscopicmatrix G(Ω) to that of the rotor Gt(Ω) as no gyroscopiceffects are present within the housing. The input andoutput matrices J, L could also be easily divided into twoparts corresponding to the rotor and the housing.


3583

Fig. 3. Schematic of the loop calculation involving thethree main sub-models

The decoupling of the overall stiffness and damping ma-trices K and D requires special attention as they expressthe coupling of the rotor to the housing through the fluidbearings. In order to preserve the overall dynamics, thematrices of the equivalent spring-damper model Do andKo are first calculated. These matrices are obtained bysubtracting the rotor and housing part Kt,Dt and Kh,Dh

from the coupling terms in the K and D matrices, basedon the layout of Eq. (4). The input and output matrices ofeach of the three sub-models are then chosen so that thevelocity and position of the interface nodes at the rotor andthe housing can be extracted, so that the force interactionswith the fluid bearing can be calculated. Such a settingallows a loop calculation as shown in Figure 3.

4.1 Mathematical model of the decoupled systems

Now, assuming that the overall model of the machine isavailable in the form (5), the decoupled systems of therotor and the housing are then,

Mtzt + (Dt + Gt(Ω))zt + Ktzt = (Jt Jt,o)

(ut

−Fo

)(

yt

yt,o

)=

(Lt

Lt,o

)zt,

(9)

and

Mhzh + Dhzh + Khzh = (Jh Jh,o)

(uh

Fo

)(

yh

yh,o

)=

(Lh

Lh,o

)zh,

(10)

where Jt,o and Jh,o are the newly inserted input matricesto account for the fluid bearing forces acting on the rotorand housing, respectively. Similarly, matrices Lt,o and Lh,o

are the newly added output matrices to allow the forcesof the rotor and that of the housing acting on the fluidbearing to be calculated using

Do(yt,o − yh,o) + Ko(yt,o − yh,o) = Fo. (11)

It is important to point out that the difference betweenthe new added outputs is included in Eq. (11), becausethe vector of the generated forces Fo is caused by therelative displacement and velocity of the spring-dampersystem representing the bearing. Caused by the fact thatthe velocity is needed, it is necessary to get over to a firstorder model (see Sec. 5). Please note also that the numberof inputs and outputs added to the sub-models dependson the interface between the bearing and the rotor fromone side and the bearing and the housing from the otherside. Hence, the number of inputs and outputs can be quitelarge once these interfaces involve a large number of nodes.

4.2 Automation of the partitioning step

As the models being considered are large scale models withseveral thousands of equations, it is almost impossible topartition their corresponding system matrices manually.Hence, there is a need for a numerically efficient algorithmthat automatically extracts the matrices of the three sub-models once those of the overall model are given. This is infact possible, once the information about the correlationbetween equation number, node number and degrees offreedom is available. Fortunately, to the author’s knowlege,all FEM software tools can deliver such information. It isgenerally in a text file where each line contains the orderedsequence of node number, degree of freedom and equationnumber. This information can be expressed as a matrixdenoted as N ∈ Nn×2, where N(i, 1) = j and N(i, 2) = krepresent the i-th equation, node number j and k-th degreeof freedom.

The automation of the decoupling of a large-scale rotordynamical system which is given in the form of Eq. (5)with N is summarized in the following algorithm:

Algorithm 1 Partitioning Algorithm

Input: M,D,G,K,J,L and N.1: Identify all equations involving a node which is affected

by a gyroscopic moment (rotor nodes) and merge theirnumbers in the set

R= i ∈ [1, n] | ∃ a, b ∈ [1, n] :

N(i, 1) = N(a, 1) ∧G(a, b) 6= 0.2: Define the set of equations involving the housing

nodes as

H = [1, n]\R.3: Knowing R and H, the mass, stiffness, gyroscopy

and damping could be separately stored for rotorMt, K

∗t , D

∗t , Gt and housing Mh, K

∗h, D

∗h.

4: Identify all elements K(i, j) 6= 0 where i is out of Rand j is out ofH. Store the identified stiffness elementsin the new bearing stiffness matrix −Ko. Merge theequation numbers in the sets

Ro = i ∈ R| ∃j ∈ H : K(i, j) 6= 0Ho = i ∈ H| ∃j ∈ R : K(i, j) 6= 0.

5: Calculate the rotor stiffness matrix Kt and that of thehousing Kh by subtracting the bearing stiffness Ko

from K∗t and K∗h, respectively.6: Generate the new inputs Jt,o, Jh,o and outputs Lt,o,

Lh,o out of the equation numbers of Ro and Ho.7: Repeat step 4, 5 and 6 for the damping matrix.

Output: The matrices of the three sub-models (9), (10),and (11).

5. ORDER REDUCTION AND RE-COUPLING THEOVERALL SYSTEM

Now that the three sub-models are on hand in a second-order form, the last step before model reduction would beto transform them into a classical first-order state-spacerepresentation of the form Eq. (6). The following step is


3584

the same for the housing and the rotor and for simplicityonly shown for the latter.

Et︷︸︸︷(I 00 Mt

) xt︷︸︸︷(ztzt

)=

At(Ω)︷︸︸︷(0 IKt (Dt + Gt(Ω))

) xt︷︸︸︷(ztzt

)

+

(Bt Bt,o)︷︸︸︷(0 0Jt Jt,o

) ut︷︸︸︷(ut

Fo

), (12)(

yt

yt,o

yt,o

)︸︷︷︸

y∗t

=

(Lt 0Lt,o 00 Lt,o

)︸︷︷︸(

Ct

Ct,o

)(ztzt

).

At this point, it should be noted that the inverse ofthe matrix Mt for large-scale systems should be avoided,which is why a matrix Et has been included in the systemrepresentation above. Moreover, even though it is not themost common way, this choice of Lt,o is needed to obtainyt,o as an output to calculate the fluid bearing force (seeEq. (11)).

The model of the rotor is always a parametric dynami-cal system with the gyroscopic matrix depending on therotational speed, as shown in Sec. 2. Thus, a parameter-preserving model reduction of this sub-model is reason-able, however, unnecessary for the class of machines con-sidered in this work, where the model of a rotor nor-mally consists of only 50–100 nodes, i. e. involving between200 to 600 state equations. The housing, in contrast, isa parameter-free system with more than 100,000 statevariables. Hence, it is advisable and advantageous to onlyreduce the housing model with a classical MOR methodand re-couple it to that of the rotor instead of reducingthe overall model with a parameter-preserving approach.

The reduction of the housing sub-model by a two-sidedKrylov-subspace method as shown in Sec.3, leads to thefollowing reduced-order model

Eredh xred

h = Aredh xred

h +(Bred

h Bredh,o

)(uh

Fo

)

y∗h =

yh

yh,o˙yh,o

=

(Cred

h

Credh,o

)xredh .

(13)

One advantage of the partitioning/re-coupling approachpresented in this work is that it connects the sub-modelsthrough their inputs and outputs, which are quantitiesthat can not be eliminated by the MOR step, unlikethe state variables. A block diagram illustrating the re-coupling of the overall system using the reduced housing isshown in Figure 3. To avoid the numerical and implemen-tation disadvantages occurring from loop calculations, thereduced system Eq. (13) is re-coupled to the unmodifiedmodels of the rotor and bearing. This is done by combin-ing Eqs. (9), (11) and (13). Nevertheless, the reductionintroduces a small error in yh, that propagates throughthe state vector xt to the output yt.

Fig. 4. Simplified FE-model of the gas compressor driveconsisting of: rotor (blue), bearing housing (green),the model of the fluid bearing (red) and spring-damper devices (gray).

The final result of partitioning, reducing and re-couplingis expressed by Eq. (14). Note that the inversion of thereduced matrix Ered

h and that of Mt is numerically possibleand not time consuming, as both quadratic matrices arenon-singular and only have a few hundred rows.

6. TECHNICAL EXAMPLE

To illustrate the approach of this paper, a simplified FE-model of the gas compressor drive shown in Figure 1 isconsidered. This machine, with a power of 16 MW, wasbuilt for an offshore platform in Norway. Its rotor hasa length of nearly 6 m and the housing has a quadraticbase with a length of 4 m. The simplification of the modelinvolves just considering the rotor, the fluid bearing, andtheir housings while approximating the rest of the housingusing several spring-damper-devices as shown in Figure 4.By taking the constraints of several degrees of freedominto account, the FE-modeling leads to a second ordersystem of the form Eq. (5) with 4204 states. The consideredmachine has two fluid bearings which are represented byspring-damper-devices and each assumed to be connectedto the rotor at a given node.

The resulting system is first partitioned with the proposedalgorithm 1 within 1 s, leading to the rotor system with364 states and the simplified housing system with 3840states. Both systems’ terminal matrices are augmented by4 new inputs and outputs which allows the interactionforce transmitted through the bearings to be calculated.The resulting multi-input and multi-output system ofthe housing is then transformed into a first-order systemwith 7680 states and reduced down to 540 states by theKrylov-subspace method introduced in Section 3. Theexpansion point was chosen as s0 = 56 rad/s, leading toan excellent approximation of the transfer function withinthe bandwidth of interest (see Figure 5).

In order to analyze smooth running of the machine, it isnecessary to observe the vibration amplitude at severalnodes of the housing for different rotor excitations. As therotational speed of the rotor constitutes the main vibrationsource, the bandwidth of interest for these excitations iscommonly chosen as twice the nominal rotational speedof the rotor. In the amplitude plot of a given housing


3585

(˙xt

xredh

)=

At(Ω)−

(0

M−1t Bt,o

)(Ko Do

)(Ct,o 00 Ct,o

) (0

M−1t Bt,o

)(Ko Do

)Cred

h,o

(Ered

h

)−1Bred

h,o

(Ko Do

)(Ct,o 00 Ct,o

) (Ered

h

)−1 (Ared

h −Bredh,o

(Ko Do

)Cred

h,o

)(

xt

xredh

)

+

(

0

M−1t Bt

)0

0(Ered

h

)−1Bred

h

(ut

uh

),

(yt

yh

)=

((Ct 0

)0

0 Credh

)(xt

xredh

)(14)

Fig. 5. Comparison of the original model (8408 states) andthe reduced model (1268 states); Vibration amplitudeof a housing node depending on different excitationfrequencies of the rotor node at a rotational speedΩ = 3000 rpm

node presented in Figure 5, the nominal rotational speedis chosen to be around 3000 rpm, corresponding to an ana-lyzed frequency interval of between 0 rad/s and 630 rad/s.The relative error resulting from reducing the model of thehousing and re-coupling it with the rotor is smaller than0.01 % up to 740 rad/s.

The calculation of the frequency response of the origi-nal system of the housing for a single rotational speedtakes around 121 s with optimized tools, while that ofits corresponding reduced system only takes 10 s. A sim-ulation of the complete re-coupled system including re-duction requires around 24 s. Hence, for a critical-speedmap, i. e. a simulation with 15 different rotational speeds,the complete simulation with the original model wouldneed about half an hour while with the reduced model itwould take only 174 s including the decoupling, reduction,and re-coupling steps. This corresponds to a relative timesaving of over 90 % for this relatively small problem. Whenconsidering the whole machine shown in Figure 1, firstsimulations have shown that model reduction slashes thecalculation time for a critical-speed map from about threeweeks down to just a few hours!

7. CONCLUSION

In this paper, the problem of order reduction of largerotordynamic systems occurring in industry has beenconsidered. Under the common assumption that only thecomplete model of the machine can be obtained fromthe FEM software tool, an algorithm for partitioning thelarge drive system into the rotor, housing, and bearingsub-models has been introduced. Model order reduction

has then been applied, when necessary, to each of thesesystems allowing a drastic time saving in the simulation ofthe re-coupled overall system. However, a number of openquestions has still to be solved to optimize the presentedapproach, including model reduction of systems with alarge number of inputs and outputs, the consideration ofthe nonlinear effects in the equivalent model of the fluidbearings describing the dependency of the oil film on therotational speed, and the analysis of the error dynamicswithin the re-coupled system caused by the reduction step.

REFERENCES

Antoulas, A.C. (2005). Approximation of Large-ScaleDynamical Systems. SIAM, Philadelphia.

Athavale, M.M. and Hendricks, R. (1996). A small pertu-bation cfd method for calculation of seal rotordynamiccoefficents. International Journal of Rotating Machin-ery, 2(3), 167–177.

Daniel, L., Siong, O., Lee, K., and White, J. (2004).A multiparameter moment matching model reductionapproach for generating geometrically parameterizedinterconnect performance models. IEEE Trans. onComputer-Aided Design of Integrated Circuits and Sys-tems, 5, 678–693.

Freund, R.W. (2003). Model reduction methods based onKrylov subspaces. Acta Numerica, 12, 267–319.

Friswell, M.I., Penny, J.E.T., Garvey, S.D., and Lees, A.W.(2010). Dynamics of Rotating Machines. CambridgeUniversity Press.

Glienicke, J., Han, D.C., and Leonhard, M. (1980). Prac-tical determination and use of bearing dynamic coeffi-cients. Tribology International, 297–309.

Grimme, E.J. (1997). Krylov Projection Methods for ModelReduction. PhD thesis, Dep. of Electrical Eng., Uni.Illinois at Urbana Champaign.

Huebner, K.H., Dewhirst, D.L., Smith, D.E., and Byrom,T.G. (2001). The Finite Element Method for Engineers.Wiley-Interscience.

Panzer, H., Mohring, J., Eid, R., and Lohmann, B. (2010).Parametric model order reduction by matrix interpola-tion. at – Automatisierungstechnik, 58(8), 475–484.

Peng, L., Liu, F., Pileggi, L.T., and Nassif, S.R. (2005).Modeling interconnect variability using efficient para-metric model order reduction. In Proc. of the Design,Automation and Test In Europe Conference and Exhi-bition. Munich, Germany.

Yamamoto, T. and Ishida, Y. (2001). Linear and Nonlin-ear Rotordynamics: A Modern Treatment with Applica-tions. Wiley-Interscience.

Zienkiewicz, O.C., Taylor, R.L., and Zuh, J.Z. (2005). TheFinite Element Method: Its Basics and Fundamentals.Butterworth-Heinemann.


3586

Documents

Model Order Reduction of Rotordynamical Systems · reduced using a suitable reduction method. For instance, for large-sized systems with Krylov-subspace methods. The resulting subsystems