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2) The reliability of MM and BT in terms of statics.

3) The practical significance of the error bound of BT

when the boundary conditions of the input DOF in the

reduced system divers significant from the situation at

the time of trial vector generation.

4) The reliability of the methods when large parts of a

structure are not represented by the I/O. This is of par-

ticular interest when large models like car bodies or air-

crafts with few I/O locations will be reduced.

A preliminary und unfinished version of this work has

been published at the SEM IMAC Conference in Jack-

sonville, FL, US [26].

The paper is organized as follows: In a first section

general issues connected to model reduction will be dis-

cussed as well as issues arising from the prior mentioned

topics. In the subsequent section all three approaches will

be briefly outlined as far as it is necessary to see the

qualitative differences. At the end of this chapter the

most important conclusions already can be drawn based

on the given equations. In the following a simple beam

example is used to demonstrate the expected advantages

and disadvantages of each method which leads to a clear

conclusion and recommendation. Finally some publica-

tions will be briefly discussed.

2. General Issues on Model Reduction of

Metallic Structures

The FE method leads to an equation of motion in the

form of

1

1

Mx KxBu

yCx (1)

where the (n × n) matrixes M and K are the mass and

stiffness matrix, the (n × 1) vector x contains the bodies

DOF, the (u × 1) vector u holds the time history of the

applied loads, the (n × u) matrix B1 maps the loads to the

corresponding degrees of freedom, the (y × 1) vector y

holds the output of interest and C1 maps the state vari-

ables to the output. In the following a symmetric system

is considered with M = MT and K = KT. In next subsec-

tion it will be discussed why no viscous damping is con-

sidered for the reduction process.

The corresponding reduced system can be given as

1

1

1111

,,,

,

TT

T

MzKzB u

yCz

xVzMWMV KWKV

B=W BC=CV

(2)

with the (m × 1) vector z and the (n × m) matrixes V and

W. Model order reduction deals with the question of de-

termining the matrixes V and W, so that the reduced sys-

tem captures somehow the important characteristics of

the unreduced system.

For a better readability of this work it is remembered

that (1) can be transformed into a first order system in the

form of

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313

1

1

,,

,

TTT

00

00

00

EqAq Bu

yCq

qxxB CC

B

II

EA

MK

(3)

In [15] a little modification of (3) is suggested, so that

symmetric system matrixes E and A are obtained:

,

00

00

E

qAqBu

yCq

K

K

EA

MK

(4)

The latter modification has some important implica-

tions which will be reported in subsection 2.2. The ac-

cording reduced system is

,,

,

TT

T

EqAq Bu

yCq

qVqEWEV AWAV

BWBCCV

(5)

with the (2n × m) reduction matrixes V and W.

2.1. Absence of the Viscose Damping Matrix

The damping matrix is not considered in Equation (1)

because of the nature of the two most frequently involved

dissipation mechanisms, namely material damping and

joint damping (micro slip). Both of them are energy dis-

sipation mechanisms which cannot be properly modeled

by pure viscous damping. Material damping of metallic

structures will be overestimated in case of high fre-

quency if it is modeled with a viscous damping matrix,

see [27-29] for an experimental verification with alumi-

num. The reason why material damping is commonly

modeled with a velocity proportional approach is more

its simplicity than its accuracy. Furthermore it is mention

worth that material damping of metallic structures is that

low so that it is normally dominated by other damping

mechanisms like joint damping or by external applied

damping like rubber bearings, lubrication bearings or the

like. Joint damping (micro slip) is frequently independent

and cannot be modeled by a viscous damping at all, see

exemplarily [30]. Consequently, a viscous damping is

always an inaccurate approximation for metallic struc-

tures and it does not matter if it is introduced in the full

system or in the reduced system. When the damping ma-

trix is regarded in the reduction process, it will influence

the final comparison between the reduced and the full

model. This doesn’t make sense because something physi-

cally not meaningful is introduced and may affect an

objective evaluation. Note, that the latter considerations

may not be valid for structures with a large amount of

rubber, plastic or other visco-elastic materials.

2.2. Stability and Symmetry of the Reduced

System

If the reduced model (2) is used for time integration it is

of significant importance, that no instability will be in-

troduced due to the model reduction. In this particular

case, when no damping is regarded the eigenvalues of the

reduced model are required to be real, such as they are in

the original system. The symmetry of a mechanical sys-

tem is a fundamental characteristic which should be pre-

served during the reduction process.

Both is obtained in case of a so called “symmetric or

single sided projection” which requires an identical re-

duction and test space matrix

WV (6)

In case of a symmetric system (4) and when the input

DOF are the same as the output DOF

11

TCB it has

been demonstrated in [7] and [15], that MM and BT

leads to a symmetric projection (6).

Note, that the latter, more mathematical considerations

do have a mechanical pendant. In case of a coordinate

transformation it is common in mechanics to apply the

principle of virtual work, which requires, that the work of

all forces due to a virtual displacement has to vanish.

This can be written as

1

T

0

xMxKxBu (7)

where δ indicates a virtual displacement of the state vec-

tor. If the reduction law x = Vz is used for the state vector

as well as for the virtual displacement the reduced system

(2) is obtained with W = V. For the latter reasons we re-

strict ourselves to such systems where 11

TCB.

3. Brief Review on Component Mode

Synthesis, Moment Matching and

Balanced Truncation

For the sake of simplicity and readability an invertible

stiffness and mass matrix will be assumed further one.

The presence of rigid body modes doesn’t have an effect

on the conclusions which will be drawn. As mentioned

before, the theory given next is by far not exhaustive but

should give at least as much insight, that the finally

drawn conclusions are understandable.

3.1. Component Mode Synthesis, See [4-6]

A CMS based reduction is typically directly applied to

the second order system (1). Commonly, the reduction

matrix is of the form

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314

SD

VVV

(8)

where the trial vectors in the (n × p) matrix VD are vibra-

tion modes which capture the systems dynamics. These

modes are obtained by a proper eigenvalue problem

which delivers the modes eigenfrequencies. For the re-

duction process just those p modes (p << n) are regarded,

having an eigenfrequency smaller as a defined limit,

which typically a factor of 1.5 - 2 higher as the highest

relevant frequency content of all involved forces.

In order to improve the convergence, some static trial

vectors are added to the vibration modes. These trial

vectors are obtained by loads on the input/output DOF

and collected in the columns of the (n × s) matrix VS.

During the last decades a lot of methods for the com-

putation of V according to (8) have been suggested. An

overview is given by Craig in [5]. In the latter publica-

tion it has been shown, that the different approaches span

a similar space. For this publication the most frequently

used method has been chosen to represent the family of

“mode based” reduction methods, namely the fixed

boundary CMS introduced by Craig (and Bampton) [31].

According to Craig’s method the vector of nodal DOF

of the FE model is subdivided into

TTT

B

I

xxx

(9)

where the (b × 1) vector xB represents the “boundary”

DOF and the

1nb vector xI holds the remaining‚

inner DOF. The boundary DOF is those DOF on which

external forces may be applied. In this work these DOF

are mostly denoted as input/output DOF. In terms of (1),

the input matrix B1 contains non zero entries at these

DOF. According to the subdivision (9), an eigenvalue

problem in the form of

2

,,,IIiII Di

0KMv

(10)

can be used for the computation of the vibration modes.

The

nb nb matrixes KI,I and MI,I denote the

portion of K and M matching the definition of (9). The

obtained modes vD,i are named as ‚Fixed Boundary Nor-

mal Modes’ and can be given as

,1 ,2,

D

D

DDp

0000

Vvv v (11)

with an user defined eigenfrequency limit ω* so that

p

(12)

As mentioned before, the latter limit is usually con-

nected to the highest relevant frequency content of the

excitation. Die Matrix VS is obtained according to a

Guyan [32] reduction

1

,,

S

I

IIB

I

VKK (13)

with the (b × b) identity matrix I. Obviously, each boun-

dary DOF introduces a trial vector in VS.

It is known that the use of such static displacement

fields significantly improve the convergence of the solu-

tion in case of imposed boundary conditions which are

different to the one at the time of mode generation, see

reference [24]. Therefore it is to expect, that CMS should

deliver good results independently of the applied bound-

ary conditions. This is an important feature for accurate

stress recovery as well, see [24].

Unfortunately no “a priori” error bound in terms of

displacement is known yet. However, the procedure en-

sures exact statics and the presence of all modes of inter-

est within a well defined frequency range.

3.2. Moment Matching, See [7-11]

With the Laplace transformation the transfer function of

the system (4) can be given as

1

ss

H

CEA B

(14)

where s holds the complex Laplace variable. An ap-

proximation of (14) with a Power Series around s0 gives

0

0

0

j

s

j

j

s

ss

HT (15)

with the so called j-th moment.

011

00

j

s

jss

TCEAEEAB (16)

see [7]. If W and V are chosen in such a way that the

spanned space is equal to

110

11

110

11

m

m

vv vvv

m

m

WW WWW

Vv vv

PR PRR

Ww ww

P

RPRR

(17)

with

1

0

1

0

0

0

v

v

TT

W

TT

v

s

s

s

s

REAB

PEAE

REAC

PEAE

(18)

it can be shown, that the first 2 m moments of the re-

duced system around s0 are equal to those in the full sys-

tem.

00 1,, 2

ss

jj jm

TT (19)

Note, that a direct implementation of (17) for the con-

struction of W and V is numerically disadvantageous.

The literature offers better choices, see exemplarily [11].

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315

The trial vectors of (17) are also called “Krylov se-

quences”. When M = MT, K = KT and CT = B it easily can

be seen, that W evaluates exactly to V, see (17) and (18).

The transfer function of the second order system (1) is

1

2

11

ss

H

CM KB (20)

The formal similarity of (20) and (14) can be used to

obtain the desired quantities in a similar matter. Again W

evaluates to V in case of a symmetric system (M = MT, K

= KT) and when the input is equal to the output

11

TCB.

110

11

1

2

01

1

2

0

m

m

s

s

Vv vv

PR PRR

RMKB

PMKM

(21)

The computation of V in the presence of a damping

matrix can be found in [7].

A more mechanical interpretation of the Krylov se-

quences around s0 = 0 will be given next and can be

found in [33,34]. There, each additional Krylov sequence

regards the dynamic residua of the already existing Kry-

lov subspace in a quasi-static matter. In a first step the

state vector of (1) is approximated by the static only.

This leads to

11

1

10

xKBux vux (22)

where x(1) is the (dynamic) residuum of this quasi static

approximation and K–1B1 is selected as first trial vector v0

which is scaled by u. In a next step the acceleration vec-

tor of (1) is replaced by the second derivative of (22)

with respect to the time. This gives

11 1

1

MxKxMKBu (23)

As before x(1) is approximated by the static response in

the form

122

11

11

xKMKBuxvux (24)

where x(2) is the remaining (dynamic) residuum and the

second trial vector v1 is introduced. Substituting (24) into

(22) delivers

2

01

xvuvux (25)

and it is obvious that the latter procedure can be repeated

as long as the result has the desired accuracy. The ob-

tained vectors v0, v1, ··· can be collected in a matrix V.

They are identical with the one of (21) in case of s0 = 0.

A characteristic of a reduction base formed of Krylov

sequences is the presence of the structures static response

v0. With the same considerations as before it can be ex-

pected, which varying boundary conditions may not lead

to unacceptable results. Again there is no “a priori” error

bound available and furthermore no “a priori” frequency

limit for the validity of the reduction base can be given.

This is a drawback when it comes to the question how

many trial vectors have to be selected “a priori” in order

to get accurate results.

3.3. Balanced Truncation, See [9,10,12-15]

For the introduction of the Gramian matrixes the system

(3) is considered with E = I. The conversion into such a

system is trivial in case of a non singular E. For the sys-

tem (3) an input signal u (t) can be given, so that each

arbitrary state q (T) can be reached within the time T

starting at q0 = 0. This input signal depends on the so

called Gramian controllability matrix WC (T) and can be

given as

1

ee

T

Tt T

C

tTT

AA

uB Wq (26)

with

0

eed

T

T

tT t

CTt

AA

WBB

(27)

The important point for the issue under consideration

is that the inputs are somehow connected with the states

via the Gramian controllability matrix. Further one the

L2-norm for u (t) is

21

2

T

C

tT TT

uqWq (28)

The Gramian observability matrix WO (T) can be un-

derstood in a similar way. It can be used to reconstruct

the initial condition q0 based on an arbitrary output y (t).

This can be given as

0t

u

1

0

0

ed

T

T

tT

OTtt

A

qWCy (29)

with

T

0

eed

T

tT t

OTtAA

WCC

(30)

and the L2-norm of y (t) is

2

00

2

T

O

tTyqWq (31)

Note, that full controllability and full observability

have been assumed for the latter considerations. The par-

ticular Gramians WC (∞) und WO (∞) fulfill the Lyapunov

equation

TT

CC

TT

OO

0

0

AWWABB

AWWAC C

(32)

In case of CT = B and a symmetric system of the form

(4), the matrixes WC (∞) und WO (∞) are identical and

can be computed by the generalized Lyapunov equation

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316

TT

T

CC

0ΕWA AWΕBB (33)

see [15]. At this point it is important to note, that system

(4) und its Gramians are not unique. With each non sin-

gular (2 n × 2 n) matrix T* a transformation in the form

of

qTq (34)

can be performed which leads to an equivalent but dif-

ferent system. A system is called “balanced” when the

Gramians are diagonal and WC = WO. The corresponding

transformation is unique and the transformation matrix

will be denoted as T. For the balanced system, Equation

(28) evaluates to

1

2

2

2

2

1

1

1

1

T

n

n

ii

ii

tTT

qT qT

uq q

(35)

with σ1 > σ2 > ··· > σ2n. Equation (35) gives insight how

the energy of the input signal is distributed among the

states which correspond to the vectors of the transforma-

tion matrix T. A small σj leads to a large 1

j

and that

means that the vector j needs a lot of energy from the

input signal to be controlled. Equation (31) can be writ-

ten as

1

2

00

2

2

2

0, 0,

1

T

n

n

ii i

i

t

qq

yq q

(36)

and it gives insight how the energy of the output signal

depends from the. A large value of σj means, that the

energy of the output signal is strongly influenced by this

particular state.

The idea of BT is, to use just these states which need

low energy to be controlled from the input and which

give a lot of energy to the output. Note, that the values of

σj decrease very quickly for common mechanical struc-

tures so that just a view columns of the transformation

matrix need to be considered as trial vectors for model

reduction. A special feature of this approach is an “a pri-

ori” error bound in the form of

22

δBT u

yy (37)

with

2

1

2

n

B

Tj

jm

(38)

and m as the number of the last considered trial vector,

see exemplarily [10,20,21].

Note, that the upper error bound (38) is just valid for a

balanced realization of a first order LTI system in the

form of (4) or (3) and does not hold when BT is direct

applied to second order systems (1). A direct application

of BT on second order systems is somehow different.

Instead of single controllability and observability Gramian

matrixes there are Gramians for the positions and veloci-

ties. The interested reader is referred to [6,13] for a de-

tailed discussion and more literature quotations. In [9,16]

a modified error bound has been developed for such sys-

tems in case of frequency weighted Gramians. For more

literature on actual implementations of BT for first order

LTI systems and second order systems see [9,10,12-15].

In [35] the use of MM together with BT has been sug-

gested. MM is used there in order to transform a huge

system into one of moderate size. In a subsequent step

BT has been applied in order to transform the moderate

system into one of small size.

The Gramians obtained by the Lyaponov equations

consider all frequencies but a direct solution can be very

expensive for large systems. It is possible to compute or

approximate the Gramians for a frequency range of in-

terest, see [9] or [20]. A low rank approximation of the

solution of the Lyaponov equation can be found in [36].

One approach for the reduction of second order sys-

tems with an approximated position controllability Gramian

is reported in [9] and given next. It delivers valuable

qualitative insight into the nature of BT, especially in the

framework of un- or lightly damped structures. Similar

considerations can be found in [20] for arbitrary first

order LTI systems.

In a first step a series of frequency response computa-

tions of (1) for k frequencies in the form of

2

1 Limit

01

iii ik

MKX B (39)

have to be carried out. The resulting (n × u) deformation

matrixes Xi are collected in the

nku matrix

2ik

X

XX X. (40)

In a subsequent step the matrix X is decomposed in its

Proper Orthogonal Modes (POM) by using Proper Or-

thogonal Decomposition (POD) which is also known as

Karhunen-Leove decomposition. The interested readers

are referred to [37,38] for a short review on the POD

itself and on the available literature. In [39] a detailed

outline of the theoretical background can be found. Due

to the definition of POD the first m POM v1 ··· vm can be

characterized as

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317

1

2

2

11 1

,,

2

2

1

max

..

n

m

TT

kxu

TT

mm

kxu

T

ji ij

st

uu

xvx v

xvx v

vv

(41)

where δij denotes Kroneckers delta. The vectors v1 ··· vm

can be interpreted as those which span “most” of the

space of X in a Euclidean sense. Note that the POM are

the eigenvectors of XXT and the magnitude of the ac-

cording eigenvalues correlate with the importance of a

POM, analog the entries σi of (35) and (36). For a lot of

mechanical systems these eigenvalues decrease rapidly

so that m << n. Finally the vectors v1 ··· vm will be used

for the reduction matrix V. As observed in [9] the matrix

XXT is an approximation of the position controllability

Gramian which is the link of this approach to BT.

In [40] it has been demonstrated, that the POM of an

undamped system with a mass matrix proportional to the

identity matrix are identical with the free vibration

modes. This observation can be generalized to systems

where M and K fulfill the criteria given in [41]. The latter

publication focuses on necessary and sufficient condi-

tions so that the vibration modes are orthogonal with

respect to the mass and stiffness matrix and to them-

selves. Note, that the vibration modes orthogonality with

respect to themselves is not true in general.

For the following considerations it is assumed that the

number of frequency samples k off (39) is high enough

so that each reachable vibration mode will be excited

nearby its eigenfrequency. Therefore each excited eigen-

frequency will lead to huge contributions in X. Now it

can be concluded that the space spanned by X is domi-

nated by these (linear independed) vibration modes. The

later considerations and (41) require that the first and

most important POM approximate the space spanned by

the dominating vibration modes.

The fact that the conservative system may have infi-

nite response, implies that the “a priori” error estimation

(37) is just valid in presence of a viscous damping which

is not realistic in the framework of metallic structures,

see the remarks in chapter 2.1. This is the first observa-

tion why the “error controlled model reduction” is more a

theoretical statement than a practical one, when applied

to the reduction of metallic structures.

The static response of the structure is undetermined

because in general it cannot be ensured that it can be ap-

proximated by a space spanned by the vibration modes.

Especially when the static response in a dynamical stiff

direction is needed, the displacement error may be small

but the error in terms of stress is not predictable which is

demonstrated in the numerical example below. Note that

in a dynamic response computation the exact representa-

tion of a static portion is very important because it leads

to mean stress which is a significant parameter for fa-

tigue lifetime prediction.

In terms of dynamics, it is known that static deflection

shapes, which help to fulfill arbitrary boundary condi-

tions, do accelerate the convergence of a mode base [24].

Therefore it is to expect, that a balanced reduced system

does not deliver accurate results in case of mounted or

mistuned I/O DOF. The observation that the absence of

static displacement fields may lead to considerable errors

can be found in [42] as well. This publication deals with

the application of POD to linear dynamic systems. The

relation to BT is not mentioned there. However, a sug-

gestion is given to improve the reduction base by a static

correction.

Finally it is important to emphasize the difference be-

tween the eigenfrequencies of (10) and the singular val-

ues (SV) σi of (35) and (36). An eigenfrequency has a

physical meaning and denotes a pole in the system char-

acterized by M and K. The SV describes the importance

of a trial vector in terms of a Euclidean distance or an

energy transfer from the input to the output. This differ-

ence leads to two interesting facts:

A certain mode will not be detected by BT when the

locations of the input/output DOF are at this modes

vibration node.

The sequence of the POM is not directly connected

with the eigenfrequencies of the structure but with the

location of the input/output DOF. That means that

there is no direct correlation between the number of

trial vectors and spanned frequency range.

3.4. Short Remark on the Number of Input/

Output DOF

As it can be seen in the theory outlined before, the num-

ber of I/O DOF is kind of a bottle neck. Too many of

such DOF would lead to many columns in the reduction

matrixes and therefore to an undesired size of the re-

duced model. Model reduction for systems with many

input/output DOF is still an active field of research and

the interested reader is referred to [43-45] for some dif-

ferent approaches to overcome that problem.

3.5. Summary and Conclusions Based on the

Nature of the Three Approaches

1) Using CMS in a proper way leads to a reduced sys-

tem which is exact in terms of static and sufficient for a

defined range for frequency. It is to expect that the static

mode shapes will guarantee satisfying results even if the

I/O DOF is different mounted as at the time of mode gen-

eration.

2) MM leads to an exact static behavior whereas the

dynamics is not clearly predictable because no frequency

range of validity can be given for a certain number of

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318

trial vectors.

3) It is to expect, that the strength of MM and BT is

the representation of the I/O characteristic of the full

system in the reduced space. The error is a minimum and

even predictable in case of BT when no statics is of sig-

nificance.

4) For BT it can be expected that the static response is

in general not accurate. The dynamic response is not ac-

curate for a reduced system with different boundary con-

ditions as they have been at the time of mode generation.

4. Numerical Example (Cantilever Beam)

In the latter section some conclusions have been drawn

just by considering the characteristics of the reduction

methods. A generic beam example will be investigated

for a practical demonstration of the former insights.

As mentioned before a non floating structure (invert-

ible stiffness matrix) is considered in order to keep the

formalism as simple as possible without losing generality.

There is no single conclusion which would not hold in

case of a floating structure.

The beam seems a bit artificial, especially the over-

hanging sub beam. The intention was the presence of a

significant part which is not covered by the input/output

DOF. This is of practical relevance for the model reduc-

tion of large structures like car bodies or aircrafts. In

such cases it cannot be garneted in general, that the criti-

cal regions which are not known “a priori” are well rep-

resented by the I/O.

4.1. FE Model and Investigated

Figure 1 contains a visual representation of the FE model

of the beam structure. The beam has been modeled using

the CBEAM1 elements of the FE code MSC.NASTRAN

[46]. The square of the bottom section is of 100 mm by

Figure 1. Solid and wireframe representation of the FE

model under consideration.

10 mm and the one of the overhanging beam is 100 mm

by 5 mm, the Young’s modulus is of 210,000 N/mm2, the

shear modulus is of 80,000 N/mm2 and the density is of

7800 kg/m3. The structure is mounted at the grid point

where the coordinate system is located and 6 input/out-

put DOF are located at the free end of the beam which is

denoted as A and marked in Figure 1 by a circle. One

additional input/output DOF in z direction is located at

the center of the beam which is denoted as B and marked

in Figure 1 as well. Note, that the sub beam which has

its main extension in x direction will be denoted as “bot-

tom section” further one.

The mass and stiffness matrixes have been assembled

by MSC.NASTRAN [46] and exported to an ASCII File.

The content of this ASCII File has been imported to

Scilab [47] which was used for all subsequent computa-

tions.

For the evaluation of the reduction methods, the re-

duced model will be compared to the full model with

respect to statics and dynamics. In order to evaluate the

statics, simply the static deflection shapes due to certain

loads is computed in the full and in the reduced system.

The dynamics is evaluated by a comparison of the ei-

genvectors and eigenvalues of the full and the reduced

system.

Three different variants will be investigated. Note, that

the reduction base is equal for all of them. The changes

are applied at the reduced model only.

Variant 1: No additional boundary conditions are ap-

plied in the reduced system. Consequently the system is

equal to the one at the time of mode (or trial vector) gen-

eration.

Variant 2: All 6 DOF of point A will be fixed to ground.

Variant 3: All translational and rotational DOF of point

A will be fixed to ground except the translation in x di-

rection. This is similar to a “sliding” joint. An additional

mass is mounted at point A.

4.2. Error Measures

The static results will be compared by plotting the bend-

ing lines in the direction of interest. The dynamics of the

reduced system will be evaluated by means of the rela-

tive error of the frequencies and the MAC value of the

normalized mode shapes. The error in frequency of mode

number i is measured as

ii

i

i

f

f

f

(42)

and the MAC value of the corresponding mode shapes is

defined as

2

T

ii

iTT

ii ii

(43)

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where βi has a value between 0 and 1 and 1 means full

identity, see [48]. Figure 2 helps to get an impression

what βi = 0.95 means in terms of displacements. For that

reason, a value of βi < 0.95 will be denoted as inaccurate

further one.

4.3. Model Reduction

4.3.1. Moment Matching

As it can be seen in (21), moment matching leads to

blocks which are added to the reduction matrix. Due to

the 7 selected I/O DOF the number of trial vectors in the

reduction matrix is an even multiple of 7. For this study a

total number of 14 trial vectors have been selected. The

trial vectors have been computed along the algorithm

cited in [11].

4.3.2. CMS

The Craig-Bampton approach [31] requires 7 static dis-

placement vectors and a number of vibration modes which

can be estimated by the frequency content of the excita-

tion. As mentioned before, the final size of the entire

reduction matrix is given by the MM approach which

leads to 7 fixed interface vibration modes which are

added to the CMS reduction base. The highest considered

frequency is about 331 Hz. Therefore, as an “a priori”

estimation, the eigenvalue and eigenvectors should be

accurate up to approx. 200 Hz.

4.3.3. BT

The trial vectors based on balanced truncation have been

computed along the SBPOR algorithm of [15]. This al-

gorithm requires the presence of viscous damping. There-

fore a damping matrix D = 1.0e–4K has been defined in

order to obtain the trial vectors.

In Chapter 3.3, it has been assumed, that in case of full

controllability the space spanned by balanced truncation

is somehow similar to one spanned by the systems vibra-

tion modes. For a numerical confirmation of that as-

sumption the first three vibration modes of the structure

have been put into a matrix. A singular value decomposi-

tion of this matrix delivers the Hankel singular values

(HSV) 2.6, 2.2 and 1.4. In a next step the first three trial

vectors obtained by the SBPOR algorithm have been

added to the matrix which contains the three vibration

modes. The HSV of this matrix are 2.7, 2.4, 1.7, 0.006,

0.0001 and 0.0000002. The sudden decrease of HSV

reveals that this 6 dimensional space is dominated by

three dimensions. Consequently, both groups of modes

Figure 2. Two mode shapes with βi = 0.95.

(or trial vectors) span a similar space.

4.4. Variant 1: Reduced System with the Same

Boundary Conditions as at the Time of Mode

Generation

4.4.1. Stati c Response

Two different static load cases will be investigated.

Load Case 1 (LC 1): Two forces in z direction have

been applied. One about 50 N at point B and another one

about –15 N at point A.

Load Case 2 (LC 2): A single force in x direction

about 1000 N has been applied at point A.

The resulting deflection in z direction of the bottom

section due to LC 1 can be seen in Figure 3. As it was

expected the CMS and MM reduced model deliver accu-

rate results while BT does not.

The resulting deflection in x direction of the bottom

section due to LC 2 can be seen in Figure 4. This is a

very interesting result. It underlines the advantage of

static displacement shapes as trial vectors. While the

CMS and MM results are exact the BT result is highly

inaccurate. This is, because no trial vector with a domi-

nate displacement in x direction is part of the reduction

base. The reason for this is the displacement oriented

reduction procedure where directions with high dynamic

stiffness are considered to be less important, see (41).

The latter observation has important consequences for

the dynamoics as well. If an arbitrary dynamic response

contains a quasi static portion of the latter deflection in x

direction, the resulting stress state will be inaccurate even

if the displacement error may be small. The missing

mean stress would lead to bad fatigue lifetime prediction

as well.

4.4.2. Dynamics

Figure 5 contains a summary of the eigenfrequency errors

and the correlations of the according mode shapes, re-

spectively. It can be seen that up to 200 Hz, which is the

“a-priori” assumed range of validity for CMS, all meth-

Figure 3. z-deflection of bottom section due to LC 1.

Figure 4. x-deflection of bottom section due to LC 2.

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Figure 5. Error in mode frequencies and shape s.

ods deliver acceptable results except MM which leads to

a considerable error for the fifth mode at 129.9 Hz. BT

gives the most accurate results.

The frequency of mode number 7 is beyond the fre-

quency limit of 200 Hz. However, this mode is not cap-

tured by the BT reduction base at all. Both alternative

methods provide that mode in the reduced model even

the frequency error is considerable. Figure 6 contains a

visualization of mode number 7 which is dominated by

torsion of the bottom section. It can be observed in the

trial vectors φi that the magnitudes of the rotations are

much smaller as the ones of the translations. This is re-

lated to the chosen units and very common in conven-

tional FE models. The dominating deformations in terms

of magnitude are in the overhanging part which is not

represented in the input/output which are marked by two

circles in Figure 6. This observation leads to the as-

sumption that BT is somehow unity sensitive. This can

be easily observed when the relation of BT and POD is

considered, see Equation (41). There it can be seen that

the importance of trial vectors are based on a Euclidian

distance. Therefore the units do play an important role in

case of trial vectors which are dominated by rotational

part of the I/O DOF only. The absence of this mode will

have significant influence on the results of variant 2. An-

other explanation could be, that the units of the rotations

are not in the same scale as the translations.

4.5. Variant 2: Fixed End

4.5.1. Stati c Response

One static load case is considered where a single force

with 10 N is acting in z direction of point B. Figure 7 con-

tains the deflection in z-direction of the bottom section of

the beam. It can be observed, that BT leads to significant

errors while CMS and MM deliver accurate results.

Figure 6. Mode number 7 (torsion oft the bottom section).

Figure 7. z-deflection of bottom section due to a static load.

4.5.2. Dynamics

Figure 8 contains a summary of the eigenfrequency errors

and the correlations of the according mode shapes, re-

spectively. It can be seen that up to 200 Hz CMS gives

excellent because the investigated modes are part of the

reduction base, see (10). MM does not deliver that good

performance and the frequency error for the 4th mode is

remarkable even it is clear beneath 200 Hz. BT delivers

the worst result. This is because the mounted system is

badly represented in the mode base without static dis-

placement shapes. Furthermore it can be seen, that mode

number 3 is not part of the mode base at all. This is the

first torsion mode and the reasons for its absence can be

found in the previous subsection. Note, that the missing

mode could be excited by an imposed rotational motion

at point A. The absence of a mode is always a critical

issue if the full system response is of interest.

4.6. Variant 3: Large Mass Coupling

In this configuration just 5 DOF of point A will be fixed

to ground. The DOF which is associated with the transla-

tion in x direction is free and a large point mass about

250 kg is mounted at point A. The error in mode shapes

and frequencies can be seen in Figure 9.

Due to the large mass an additional mode comes into

the spectrum of interest. This mode has an eigenfre-

quency of 145.1 Hz and represents an elongation of the

bottom section along the x direction together with a

bending of the overhanging beam, see Figure 10.

It can be observed, that the torsion mode at 98.38Hz

and the latter mentioned mode at 145.1 Hz cannot be

represented by the BT trial vectors.

Note, that this is not a pure artificial construction. An

example would be a flexible slider crank mechanism in a

multi body dynamic system (MBDS). The FE model of

the elastic slider crank (or con rod) is typically modeled

without the mass which will be attached in the MBDS by

means of constraints. This mass attachment will influ-

ence the modes of the slider crank.

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Figure 8. Error in mode frequencies and shape s.

Figure 9. Error in mode frequencies and shape s.

Figure 10. Mode at 145 Hz (grey represents the reference

configuration).

5. Comments on Former Publications

In some recent publications MM and BT have been sug-

gested as the somehow better reduction methods in gen-

eral and for the use in multi body system dynamics in

particular. This is in contrast to some of the observations

in this work. This will be commented in this section.

5.1. Frequency Response as Relevant Criterion

In [16-19] the error in the frequency response functions

has been evaluated. This has been done by the compari-

son of the reduced models frequency responses with the

ones of the full system. All comparisons have in common

that the error of BT and MM are much smaller as in case

of mode based methods.

Firstly, the evaluation of the frequency response func-

tions do not properly take into account the full system

response and boundary conditions which are different as

they have been at the time of trial vector generation. The

observation in the latter publication matches Figure 5

which holds the eigenfrequencies and mode shape errors

of the reduced system. BT and partially MM deliver the

best results because the reduced system is not mistuned

at the I/O. As depicted in the sections before a deduction

off similar accuracy in case of different boundary condi-

tions is not possible. Further one it could be demon-

strated, that the quality of static response is not predict-

able.

In [18] the mode based methods are represented by the

fixed boundary CMS as in this work. The error of this

approach is less than 1% for the frequency range of va-

lidity (up to 720 Hz). MM and BT deliver better results

indeed but the result quality of CMS seems to be suffi-

cient for standard mechanical applications.

The mode based methods in [16,17,19] are not identi-

cal with a CMS reduction base in the form of (8). There-

fore the error, which goes up to 10% in the frequency

range of interest, cannot be commented here.

5.2. Time Domain Response as Relevant

Criterion

In [18] a crankshaft is reduced by MM, BT and fixed

boundary CMS. A multi body simulation of the full en-

gine fired by gas pressure has been performed. The rela-

tive error of the gap and the pressure for one particular

lubrication bearing has been evaluated with respect to the

time. A close look to the diagrams reveals, that the CMS

reduced model has the best performance in an averaged

sense. This has to be expected because an operating

crankshaft is closer to a mounted crankshaft as to an un-

supported one. The conclusion in the latter publication

has not been based on this observation but on the fre-

quency response error which is problematic as explained

in the subsection before.

6. Conclusions

This paper is devoted to a clear answer which method

has to be used for the model reduction of metallic struc-

tures in context of industrial use. The following conclu-

sions have been drawn by a close look to the underlying

theory and they have been demonstrated by means of a

generic beam example.

The clear result of the qualitative and quantitative in-

vestigations is that modal reduction techniques based on

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component mode synthesis (CMS) have the best overall

performance. The results are sufficient accurate in terms

of statics and dynamics, independently of the applied

boundary conditions. Another practical feature is a clear

selection criterion for the number of considered modes as

a function of the excitations frequency content.

The observed accuracy in case of moment matching

(MM) cannot compete with one obtained by the CMS

method. The absence of a correlation between the con-

sidered modes and the covered frequency range is an-

other drawback for industrial use where no convergence

analysis with respect of the number of trial vectors will

be done.

Finally it can be reported, that BT in principle cannot

be recommended for model reduction in the investigated

framework at all. The reasons are manifold:

If the overall system response contains a static portion,

the accuracy in terms of displacement and stress is ques-

tionable.

Due to the absence of static correction vectors in the

mode base, BT delivers bad results when the I/O DOF in

the reduced system face different boundary conditions as

at the time of trial vector generation.

Vibration modes can be missed as a consequence of

the I/O focused approach and its unity sensitivity. This

can lead to unpredictable full system response in case of

varying boundary conditions in the reduced system.

There is no correlation between the trial vectors and

the covered frequency range.

As a final conclusion it can clearly be stated that the

most reliable reduction method for metallic structures

and for a wide range of industrial application in me-

chanical engineering is still the Component Mode Syn-

thesis and its variants. Balanced truncation can be rec-

ommended, when the reduced system is not mistuned,

when the influence of statics is clarified and when just

the I/O behavior is of interest.

One of the most important fields where model reduc-

tion of flexible bodies needs to be applied is elastic multi

body system simulation (MBSS). As already mentioned

MBSS is characterized by the importance of statics, over-

all system response and varying boundary conditions at

the I/O DOF of flexible bodies. For the reasons shown in

this paper, CMS seems to be the best choice.

The feature of predictable error in case of balanced

truncation (BT) may be valuable when the displacement

I/O behavior of the non reduced system has to be ap-

proximated. The practical significance of such an a-priori

error estimation is questionable when viscous damping is

used as an approximation of real damping, when the

boundary conditions in the reduced model differ from the

one of the trial vectors and when stresses or statics are of

particular interest.

Finally it is mentioned once again, that the focus of the

current work is on mechanical engineering. The drawn

conclusions may not be valid for other disciplines like

control or electrical engineering, where the objectives

may be somehow different.

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[12]
On the Modal and Non-Modal Model Reduction of Metallic Structures with Variable Boundary Conditions
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